## Calculus AB and Calculus BC

## CHAPTER 3 Differentiation

**Concepts and Skills**

In this chapter, you will review

• derivatives as instantaneous rates of change;

• estimating derivatives using graphs and tables;

• derivatives of basic functions;

• the product, quotient, and chain rules;

• implicit differentiation;

• derivatives of inverse functions;

• Rolle’s Theorem and the Mean Value Theorem.

In addition, BC Calculus students will review

• derivatives of parametrically defined functions;

• L’Hôpital’s Rule for evaluating limits of indeterminate forms.

### A. DEFINITION OF DERIVATIVE

At any *x* in the domain of the function *y* = *f* (*x*), the *derivative* is defined as

The function is said to be *differentiable* at every *x* for which this limit exists, and its derivative may be denoted by *f* *′*(*x*), *y* *′*, or *D*_{x}*y*. Frequently Δ*x* is replaced by *h* or some other symbol.

The derivative of *y* = *f* (*x*) at *x* = *a*, denoted by *f* *′*(*a*) or *y* *′*(*a*), may be defined as follows:

• **Difference quotient**

• **Average rate of change**

• **Instantaneous rate of change**

• **Slope of a curve**

The fraction is called the *difference quotient for* *f* *at a* and represents the *average rate of change of* *f* *from a to a + h.* Geometrically, it is the slope of the secant *PQ* to the curve *y* = *f* (*x*) through the points *P*(*a*, *f* (*a*)) and *Q*(*a* + *h*, *f* (*a* + *h*)). The limit, *f* *′*(*a*), of the difference quotient is the (*instantaneous*) *rate of change of f* at point *a*. Geometrically, the derivative *f* *′*(*a*) is the limit of the slope of secant *PQ* as *Q* approaches *P*; that is, as *h* approaches zero. This limit is the *slope of the curve at P.* The *tangent to the curve at P* is the line through *P* with this slope.

**FIGURE N3–1a**

In Figure N3–1a, *PQ* is the secant line through (*a*, *f* (*a*)) and (*a* + *h*, *f* (*a* + *h*)). The average rate of change from *a* to *a* + *h* equals which is the slope of secant *PQ*.

*PT* is the tangent to the curve at *P*. As *h* approaches zero, point *Q* approaches point *P* along the curve, *PQ* approaches *PT*, and the slope of *PQ* approaches the slope of *PT*, which equals *f* *′*(*a*).

If we replace (*a* + *h*) by *x*, in (2) above, so that *h* = *x* − *a*, we get the equivalent expression

See Figure N3–1b.

**FIGURE N3–1b**

The second derivative, denoted by *f* *″*(*x*) or or *y* *″*, is the (first) derivative of *f* *′*(*x*). Also, *f* *″*(*a*) is the second derivative of *f* (*x*) at *x* = *a*.