Chapter 3: Differentiation Rules

Welcome to Last Minute Lecture.

This free chapter overview is designed to help students review and understand key concepts.

These summaries supplement, not replace, the original textbook and may not be redistributed or resold.

For complete coverage, always consult the official text.

Welcome to the Deep Dive.

We're the place you come to get the essential insights from complex topics, fast.

Today we're jumping into calculus.

Specifically, chapter three from Stuart, Clegg, and Watson's calculus, early transcendentals.

It's all about differentiation rules.

Exactly.

Our mission here is to get you past that sometimes tedious limit definition of the derivative.

Right.

We want to equip you with the powerful tools you need to actually calculate derivatives much more easily.

And we'll uncover some pretty surprising applications along the way.

We're cutting through the complexity to get you that core knowledge.

Yeah, this chapter is really foundational.

These rules simplify things immensely, but they also connect to so many important ideas in science, engineering, economics,

you name it.

Okay, let's start simple.

Maybe the simplest rule of all.

The derivative of a constant function.

Ah, yes.

So think about a function like, say, y -ways five.

Just a flat line on a graph.

Exactly.

And what's the derivative measuring?

It's measuring the rate of change or the slope of the tangent line.

And the slope of a flat horizontal line is?

Zero.

Always zero.

If something isn't changing, its rate of change is zero.

Makes perfect sense, right?

It does.

And it saves you from having to plug f -x -e -c into that whole limit definition.

Precisely.

A nice quick win.

Okay, so from zero change, let's move to, well, changing things.

The power rule.

This one is a game changer.

Oh, absolutely.

This is your workhorse rule for anything like x raised to some power, n.

So x to the n, how does it work?

Super elegant, actually.

You just bring that power n down in front as a coefficient.

And then you reduce the original power by one.

So it becomes x to the power n one.

What's this?

So if you have just x, that's x to the power one.

Bring the one down.

Reduce the power by one to zero.

One by zero is just one.

Perfect.

And x squared.

Bring the two down.

Reduce the power to one.

So two x.

Yep.

X cubed.

Three x squared.

X to the fourth.

Four x cubed.

Okay, I see the pattern.

It's really straightforward.

It is.

And think how much calculation that saves compared to using the limit definition for say x to the fourth.

Oh, definitely.

That would be a lot of algebra.

And what's really powerful is that this rule, it's not just for nice positive whole numbers like two, three, four.

Ah, right.

You mentioned it works more broadly.

Yes.

It works for negative integer powers too.

And crucially, for fractional exponents, think roots.

Okay, so like a square root.

That's x to the power 12.

Exactly.

So apply the rule.

Bring the 12 down front.

Reduce the power by one.

So 12 one is negative is 12.

So 12 x negative 12, which is one two r's.

You got it.

What about something like y x, x, x?

Okay.

First, rewrite it using powers.

X times x 12 is by 32.

Good.

Now differentiate.

Bring down the 32.

Reduce the power by one.

So 32 one equals 12.

So it's 32 by 12 or 32 x.

Perfect.

See, incredibly versatile rule.

Okay.

Building on that, what if there's just a constant number multiplying our function like five by two?

That's the constant multiple rule.

And it's nice and simple.

The constant just kind of tags along for the ride.

So the derivative of c times f x is just c times the derivative of x.

Exactly.

So for five by two, the derivative of by two is two x.

The five just multiplies that.

So five two x is 10 x.

Okay, that makes sense.

Is there a visual for that?

Yeah.

Think about stretching a graph vertically.

Yeah.

If you multiply a function by five, you're stretching it five times taller.

Right.

So at any point, the rise part of the slope is five times bigger, but the run stays the same.

So the slope itself becomes five times steeper.

The constant just scales the derivative.

Got it.

That's intuitive.

And what about adding or subtracting functions?

Even easier.

Uh -huh.

The sum and difference rules, they work exactly like you'd hope.

Meaning?

The derivative of a sum of functions is just the sum of their individual derivatives.

And the derivative of a difference is the difference of the derivatives.

Yeah, piece by piece.

So these rules together, constant multiple, power rule, sum difference, they make differentiating any polynomial almost trivial.

Yeah, you can just go term by term.

Very neat.

So that handles the basics.

But life isn't always just polynomials added together.

What if you multiply two functions, like x times sin x?

Right.

My first instinct might be to just multiply their derivatives, but I suspect that's wrong.

You suspect correctly.

That's a common mistake.

You need a specific rule for products.

The product rule.

Okay, what's the rule?

It's often phrased as the first function times the derivative of the second.

PLUS, the second function times the derivative of the first.

First times derivative of second plus second times derivative of first.

Let's try that on.

Say fx, it deals x.

We talked about x earlier.

Good example.

So the first function is x, the second is x.

Derivative of the first x is one, derivative of the second x is just x.

Right.

Now plug into the rule.

Okay.

First, x times derivative of second x, that's zx, plus second x times derivative of first one.

That's just x.

So x plus x.

Exactly.

Which you can factor as x plus one x.

Nice.

Is there an intuitive way to understand why the product rule works like that?

There is.

Imagine a rectangle with sides u and v, where both u and v are functions, maybe of time.

So the sides are changing.

Okay.

So the area a, a, u, v is changing too.

Right.

Now if u increases by a tiny amount du and v increases by dv, how much does the area increase?

Hm.

Get a little strip added on one side with area v du and another strip on the other side, area u dv, and a tiny corner square du dv.

Exactly.

And in calculus terms, that tiny corner is negligible compared to the strips.

So the change in area da is basically v du plus u dv.

That's the product rule structure right there.

Ah, that's a great visual.

It makes sense why you have those two cross terms.

No, it's cool with that zex example.

If you keep differentiating it, find the second derivative, the third, and so on, a pattern emerges.

Really?

What happened?

The nth derivative actually turns out to be x plus any x.

It's quite remarkable.

Shows how these rules can reveal deeper structure.

Wow.

That's neat.

Okay.

So product rule for multiplication.

What about division?

Like fx, gx.

That needs the quotient rule.

And it's a bit more of a beast to remember.

Uh oh.

Hit me with it.

Okay.

It's bottom times derivative of the top minus top times derivative of the bottom, all divided by the bottom squared.

Whoa.

Okay.

Low d minus high d low over low low.

That's how I learned it.

Uh huh.

Yeah.

That mnemonic works.

Gx, fx, fx, all over gx, too.

Got it.

Seems like more room for error there.

There is.

And honestly,

sometimes it's easier not to use it.

What do you mean?

Well, if you have something like sin x by 2, you could use the quotient rule or you could rewrite it as sin x, x2, and use the product rule instead.

Ah, I see.

Sometimes rewriting at first simplifies things.

Often, yeah.

It's good to have options and choose the path of least resistance, algebraically speaking.

Makes sense.

Okay.

Let's branch out from polynomials and rational functions.

Exponentials.

We already touched on x.

Right.

The star of the show.

Derivative of x is?

It's itself.

The only function up to a constant multiple that does that.

It's because e is that unique base where the slope of the tangent line at the point 0, 1 is exactly 1.

What about other bases?

Like 2x or 10x?

Good question.

The derivative of bx is bx times the natural log of the base ln b.

Okay.

So d dx 2x equals 2x ln 2.

Precisely.

The ln b factor is essentially a correction because the base isn't the perfect base e.

Got it.

So x is fundamental.

What about trig functions?

Also fundamental.

Especially sine and cosine.

Their derivatives are beautifully simple and connected.

Derivative of sine x is?

Cos x.

And derivative of cos x.

Minus sine x.

Don't forget the minus.

Right.

Cos x goes to the sine xx and the others tan cosec cec.

They all have their own derivative rules too which you can actually derive using the quotient rule on sine and cosine like tan x equals sine x cosec.

They pop up less often but good to know they exist.

Why are these trig derivatives so important?

Think about anything that oscillates or repeats.

Like simple harmonic motion, a mass on a spring.

Its position might be described by say st equals 4 cos t.

The position s is a function of time t.

So the derivative ds dt would be the velocity.

Exactly.

Using the rule velocity vt equals 4 sin 2t.

And the derivative of velocity is acceleration.

Yep.

Differentiate again.

At equals 4 cos c.

And so just by knowing the position function and these simple rules we instantly get formulas for velocity and acceleration at any time t.

That's powerful.

You can analyze the whole motion.

Absolutely.

Yeah.

And there's a cool pattern here too.

If you keep taking derivatives of cos x, what happens?

Let's see.

Cos gar sin x sin x star x sin x dot x.

It cycles.

It cycles every four derivatives which, fun fact, relies on some fundamental limits involving sine and cosine near zero to actually prove those first derivative rules.

Limits like sin going to one.

That's the one.

And cos one going to zero.

Those are the bedrock.

Okay.

This is great.

We have rules for basics.

Products.

Quotients.

Exponentials.

Trig.

But what if functions are nested inside each other?

Like sin by 2.

Now we get to the really powerful one.

The master key, I sometimes call it.

The chain rule.

The chain rule.

Sounds important.

It is arguably the most important differentiation rule.

It's for composite function, functions inside functions, fgx.

How does it work?

I've heard it described as outside in.

That's a good way to think about it.

You differentiate the outer function, f, leaving the inner function, gx inside it.

Okay, so fgx.

Then you multiply that result by the derivative of the inner function, gx.

So fgx.

Like peeling an onion layer by layer.

Exactly like peeling an onion.

Let's try sin by 2.

The outer function is sin, the inner is by 2.

Derivative of outer sin is cos.

Keep the inside x2.

So cos x2 times the derivative of the inside, f2, which is 2x.

So cos x2, 2x.

Perfect.

That's the chain rule in action.

Does it work with that other notation, the dedex stuff?

It does, and it looks really neat.

If yfu and u equals gx, then the chain rule is del x equals duty dedex.

Oh, it looks like the dudas cancel out?

Visually, yeah.

It's not really cancellation of fractions, but it's an incredibly helpful way to remember the structure.

dedu is the outer derivative, dedex is the inner derivative.

That's very clever.

And this combines with other rules, right?

Like the power rule.

Absolutely.

That gives us the generalized power rule.

If you have gxn, its derivative is ngxn -1.

That's the regular power rule part.

Times gx, that's the chain rule part, multiplied by the derivative of the inside.

Super common, super useful.

And if you have functions nested three or four deep, like sin ra causes.

You just keep applying the chain rule.

Differentiate the outermost sin, keep the inside, cos 10x, then multiply by the derivative of that inside, which itself requires the chain rule again.

Wow.

Okay, layers upon layers.

But it's systematic and critically important in the real world.

Think about designing a smooth roller coaster track.

Smoothness.

That sounds like derivatives.

It is.

Engineers need the track's function to have continuous first and second derivatives, where different pieces connect, say, a straight ramp joining a parabolic dip.

Why second derivatives, too?

The second derivative relates to curvature, or how quickly the direction is changing.

You want that to change smoothly, too, otherwise riders feel a jerk.

Ah.

So you don't want abrupt changes in acceleration.

Exactly.

And calculating those derivatives for potentially complex track segment functions, you're definitely using the chain rule.

It ensures a smooth, safe ride.

That's a fantastic application.

Calculus making roller coasters better.

Directly.

Okay,

now, sometimes equations aren't solved nicely for y.

Like, by 2 plus y2 equals 25, that's a circle.

How do you find didx there?

That's where implicit differentiation comes in handy.

It's designed for situations where y is defined implicitly by an equation rather than explicitly as yx.

Like the circle equation, or something even nastier like the folium of Descartes by 3 plus y3 equals 6 like psi.

Solving that for y looks like a nightmare.

It is a nightmare.

The explicit solution is incredibly complex.

But finding the slope, didxx, implicit differentiation makes it manageable.

How does it work?

What's the process?

You just differentiate both sides of the equation with respect to x.

The key is, whenever you differentiate a term involving y, you treat y as a function of x.

Meaning you have to use the chain rule.

Exactly.

So the derivative of y2 with respect to x isn't just 2y, it's 2y didx.

Because y depends on x.

You multiply by the derivative of the inside function, which is y.

Precisely.

So you differentiate both sides, term by term, remembering that chain rule factor didx whenever i appears, then you'll have an equation with didx in it.

And you just solve that equation algebraically for didx?

That's the whole process.

For the circle by 2 plus y2 equals 25.

Differentiate both sides.

Minus 2x plus 2y is the didx equals 0.

Now solve for didx.

2x, 2dx, 2dx, 2 by 2y, it's miss called d.

There you go.

The slope of the tangent line of the circle at any point, xy is an equals thing.

Much easier than solving for y first, which gives two functions plus urination and differentiating those.

Definitely saves a ton of effort.

Especially for complex curves.

Immense effort.

Okay, one more specialized technique.

Logarithmic differentiation.

Logarithmic.

Sound like it involves logarithm.

It does.

It's particularly useful for really complicated functions that involve lots of products, quotients, and powers all jumbled together.

Or crucially, when you have a function where both the base and the exponent involve x, like yx raised to the power.

Bro, how do you even tackle that?

Power rule doesn't work because the base isn't constant.

Exponential rule doesn't work because the exponent isn't constant.

Exactly.

Logarithmic differentiation is the trick.

What are the steps?

First, take the natural logarithm, ln, of both sides of the equation.

So ln and y, ln xx.

Because log laws let you simplify that right side.

The exponent of x can come down front.

Ah.

So ln y, x, ln x.

That looks much nicer.

It does.

Now differentiate both sides implicitly with respect to x.

Okay, derivative of ln y is 1y dx by the chain rule.

Good.

And the right side.

Lx, ln.

That needs the product rule.

Right.

It'll be x, 1x, plus ln x, 1x.

Some simplifying needed there, but doable.

Yep.

So you have 1y, dx is the result of product rule.

And the last step is just solve for dx by multiplying both sides by y.

Exactly.

And since you know yxx, you can substitute that back in at the end if needed.

That's really clever.

It turns a horrible problem into manageable steps using logs and implicit differentiation.

It's a fantastic tool.

And interestingly, it's actually used to prove the general power rule, ddxxnxxn1, works for any real number n, not just integers or rationals.

Wow.

So it's quite fundamental, too.

It is.

Okay.

So we've amassed this amazing toolkit of rules.

Let's zoom out.

What's the big picture?

What does the derivative d dx mean?

It's the instantaneous rate of change, right?

How fast y is chaining with respect to x at a specific moment.

Yes.

Yeah.

And that concept pops up everywhere.

Yeah.

In physics, we saw a positionary velocity acceleration.

Right.

First derivative is velocity.

Second is acceleration.

It tells you how motion changes.

It also describes linear density, df dx, how mass is distributed along a rod,

or electric current, dq dt, how quickly charge flows.

What about other fields?

Chemistry?

Rate of reaction, dc dt, how fast a product concentration C is increasing, or compressibility of a gas, how its volume changes with pressure.

Biology?

Population growth rate, dd dt, how fast a population n is changing over time.

Bacteria doubling.

Also, blood flow, Poiseuille's law, relates blood velocity to the distance from the center of the vessel, and derivatives tell you how velocity changes across the vessel.

And economics.

You mentioned costs earlier.

Yes.

Marginal cost, Cx.

The rate of change of the production cost, C, as you change the number of items, x.

Critically, it approximates the cost of making just one more item.

So C500, he has $15 per item, making the 501st item cost roughly $15.

Exactly.

Maybe the actual cost is $15 .01, but the derivative gives a very quick, very useful estimate for decision making.

It's amazing how this one mathematical idea, the derivative, applies to so many different things.

That's the power of abstraction.

Velocity, density, current, reaction rates, growth, cost,

all just different flavors of the same core concept, rate of change.

And this idea of rates of change lets us model things over time, right?

Like growth.

Absolutely.

This leads to the law of natural growth or decay.

It describes situations where the rate of change of something is directly proportional to how much of it there is.

D .D .I.

deak.

So the bigger it is, the faster it grows, if D .I., or the faster it shrinks, if gay.

Precisely.

And the solutions to this simple differential equation are always exponential functions.

Mind T equals C .E .K .T.

Where have we seen that E again?

It's everywhere.

This model applies to so much.

World population growth can be approximated this way, using past data to find K and predict the future.

Radioactive decay too, right?

Half -life.

Yes.

Radioactive substances decay exponentially.

The half -life tells you how long it takes for half the substance to disappear, which is directly related to that constant K.

It's the basis for carbon -14 dating ancient artifacts.

What else?

Newton's law of cooling, how an object's temperature changes over time towards the room temperature.

The rate of cooling is proportional to the temperature difference.

And continuously compounded interest in finance, A .T., equals A0E E3.

Your money grows exponentially if interest is compounded constantly.

So that simple D .I .T.

equation has incredibly wide reach.

Immense reach.

Okay.

Related to rates of change over time.

What about related rates?

That sounds connected.

It is.

Related rates problems are about finding the rate of change of one quantity when you know the rate of change of another related quantity.

Like, if you know how fast the volume of a balloon is increasing, can you figure out how fast its radius is increasing?

Exactly.

That kind of problem.

The key is finding an equation that connects the two quantities.

Like the volume of a sphere, formula B, connects volume V and radius R.

And then what?

Then you differentiate that equation implicitly, but this time with respect to time, P.

Because both quantities are changing over time.

Ah, so you'll use the chain rule again.

Like dv dt and dre P will appear.

Precisely.

Differentiating V, 43B with respect to t, gives dv dt,

gives dv dt, evil 3F33, dv dt, or dv dt equals 4d or dt.

So if you know dv dt, how fast you're pumping air in, and the current radius R, you can solve for dre t, how fast the radius is growing.

That's the method.

Find the relation, differentiate with respect to time, plug unknowns, solve for the unknown rate.

What are some classic examples?

The inflating balloon is one.

A ladder sliding down a wall is another classic.

If the bottom slides out at a certain speed, how fast does the top slide down?

Pythagorean theorem would relate the sides there.

Water filling a conical tank, how fast is the water level rising?

Cars approaching an intersection, how fast is the distance between them changing?

A spotlight tracking a person walking, how will it work?

How fast does the light have to rotate?

Lots of geometry involved, it seems.

Often yes.

Finding that initial relating equation is usually the trickiest part.

The calculus step is often just implicit differentiation with the chain rule.

Okay, one more big application area.

Using the tangent line itself.

Linear approximation.

Right, the whole idea is that near the point of tangency, a curve and its tangent line are really, really close, almost indistinguishable if you zoom in enough.

So you can use the simpler equation of the tangent line to approximate the function's value nearby.

Exactly.

The tangent line at xA is called the linearization, Lx fA plus fa xA is just the point slope form of the line.

And this is good for estimating things like 3 .98.

Yeah.

You'd linearize fx around the nearby nice point like a 4, fx 1, 2, 6, so f4 plus 4 plus f4 by 4, 2 plus 14 by 4.

Now plug in x3 plus 14 by 4, right.

Now plug in x3 .98, L3 .98, 2 plus 14, another sort of 0, 5, and those are 0 .02.

And the actual 3 .98, it's pretty close to that.

Quick and easy approximation.

Very quick.

Of course, the further x is from A, the less accurate the approximation becomes, but for nearby points, it's great.

Are there other uses?

Oh, yeah.

In physics,

approximations like sin and cos a lot one for small angles are just linear approximations around zero.

They simplify many formulas, like for pendulums or optics.

That makes sense.

And what about differentials, dx and dx?

They're closely related.

Think of dx, the differential of x, is just a small change in x.

Then d, the differential of y, is defined as xdx.

How is that different from A, the actual change in y?

Geometrically, d is the actual rise on the curve when you move from x to x plus dx.

D, on the other hand, is the rise on the tangent line over that same change dx.

Ah.

So d is the change predicted by the linear approximation.

Exactly.

And because the tangent line is close to the curve, d is often a good approximation for E, especially when gx is small.

What are differentials used for?

Mostly for estimating errors.

If you measure a sphere's radius r with a possible error dr, which we treat as dx, you can estimate the resulting error in the calculated volume v using dv equals vr.

So dv was 4 out of dr.

It tells you how much the volume error depends on the radius error and the radius itself.

Precisely.

Helps understand the sensitivity of calculations to measurement errors.

You often look at relative error, dvv, or percentage error, too.

Okay, that covers a huge amount of ground.

Is there anything else in this chapter?

Just a brief glimpse at hyperbolic functions.

Things like sin x, hyperbolic sine,

and cosh x, hyperbolic cosine.

I've seen those.

They're defined using x and ex, right?

That's right.

Cosh x plus e by 2 and sin x, xv by 2.

They have properties remarkably similar to trig functions, hence the names.

Do they show up anywhere interesting?

Surprisingly, yes.

The shape of a hanging flexible cable or chain, like a power line, isn't a parabola.

It's actually a catenary curve described by cosh x.

They also appear in describing ocean wave velocities in other areas.

Interesting.

They have their own differentiation rules, too.

They do, very analogous to the trig rules.

ddx acx equals cosh x and ddx equals sin x, no minus sign on the cosh derivative.

Fascinating how these structures echo each other.

It really is.

Stepping back from all these rules, what we've really seen is how this concept of the derivative, this measure of instantaneous change, becomes this incredibly powerful, versatile tool once you have these rules.

It moves from a theoretical limit definition to something you can actually use across almost every field imaginable.

Exactly.

It's the language for describing change, motion, growth, rates.

Everything that isn't static.

It's pretty amazing when you think about it, how that simple core idea, rate of change, connects things as different as a planet's orbit.

The cost of making one more widget, the decay of atoms, and yeah, even how smooth your roller coaster ride feels.

All unified by the elegant machinery of differentiation.

It really makes you wonder.

What other hidden connections are out there waiting to be revealed by the tools of calculus?

Something to think about.

Definitely.

Well, thank you for joining us on this deep dive into differentiation rules.

And a huge thank you to the Last Minute Lecture Team for providing the foundation for this discussion.