Chapter 8: Further Differentiation

A function exhibits increasing behavior when its derivative remains positive throughout an interval, while decreasing behavior corresponds to negative derivative values; this distinction forms the foundation for understanding curve properties. Stationary points occur where the derivative equals zero and represent candidates for optimization; these points can be classified into three categories based on how the derivative changes across the point. Maximum turning points occur where the derivative transitions from positive to negative, creating a peak in the curve, while minimum turning points show the reverse pattern with a valley formation. Stationary points of inflection are locations where the derivative equals zero but maintains the same sign on both sides, meaning the curve continues in the same direction without forming a peak or valley. The first derivative test identifies point types by examining the sign of the derivative immediately left and right of the stationary point, providing a visual and intuitive classification method. The second derivative test offers an algebraic alternative by evaluating the second derivative at the stationary point; a negative second derivative indicates a maximum, a positive second derivative indicates a minimum, and a zero second derivative requires returning to the first derivative test for classification. Optimization problems apply these concepts to real-world contexts where students must find extreme values, such as maximizing enclosed area with limited fencing or minimizing material costs in manufacturing. These problems require expressing the objective function in terms of a single variable, differentiating, finding stationary points, and verifying the nature of the critical point. Connected rates of change extend differentiation to situations where multiple variables change simultaneously with respect to time, connected through the chain rule; students calculate how one quantity changes relative to time using the relationship dy/dt equals dy/dx multiplied by dx/dt. The reciprocal rate rule provides an alternative form for finding rates of change between variables, enabling calculations of how dimensions change in geometric shapes given information about volume or area change rates.