AP Calculus: Differentiation - Definition and Basic Rules

This chapter introduces the derivative, a foundational concept in calculus that measures the instantaneous rate of change of a function. It begins by building intuition, moving from the average rate of change over an interval (the slope of a secant line) to the instantaneous rate of change at a point (the slope of the tangent line). This idea is then formalized using the limit definition of the derivative, which you will use to calculate derivatives from first principles.

A significant part of the chapter explores the concept of differentiability. You will learn the conditions under which a derivative exists and how to identify points where a function is not differentiable, such as at corners, cusps, or vertical tangents. This involves understanding the relationship between continuity and differentiability: a function must be continuous at a point to be differentiable there, but continuity alone is not sufficient.

The focus then shifts from the theoretical definition to efficient computation. You will learn and apply the first set of differentiation rules, including the Constant Rule, Power Rule, Constant Multiple Rule, and the Sum and Difference Rule. The chapter also introduces the derivatives of the basic trigonometric functions, sine and cosine. The most common difficulty is the algebraic manipulation required for the limit definition. A strong grasp of the preceding chapter on Limits is essential for success.

Formulas for AP Calculus: Differentiation - Definition and Basic Rules

Average Rate of Change: $\frac{f(b) - f(a)}{b - a}$ The slope of the secant line between x=a and x=b.
Derivative as a Function (Limit Definition): $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ Defines the derivative function f'(x) from first principles.
Derivative at a Point (Alternate Form): $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ An alternative definition for the derivative at the point x=c.
Constant Rule: $\frac{d}{dx}(c) = 0$ The derivative of any constant c is zero.
Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$ Applies for any real number n.
Constant Multiple Rule: $\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$ A constant factor can be pulled out of the differentiation.
Sum and Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$ The derivative of a sum is the sum of the derivatives.
Derivative of Sine: $\frac{d}{dx}(\sin x) = \cos x$
Derivative of Cosine: $\frac{d}{dx}(\cos x) = -\sin x$