Differentiation rules

In the last chapter, we found derivatives with limits. That works, but it takes many steps. In this chapter, we find rules that give derivatives quickly.

With these rules, you can differentiate polynomials, products and quotients, trigonometric functions, composite functions, and exponential and logarithmic functions. Each rule comes from the limit definition, so you can see why it works.

This chapter covers the following topics:

Basic differentiation rules

The constant rule, the power rule, the constant multiple rule, and the sum and difference rules, with tangent lines, horizontal tangents, and velocity.

The product and quotient rules

How to differentiate products and quotients, why the rules work, and the power rule for negative integers.

Derivatives of trigonometric functions

The derivatives of sine, cosine, and the other four trigonometric functions, and why calculus uses radians.

The chain rule

How to differentiate composite functions, the general power rule, and how to combine the chain rule with the other rules.

Implicit differentiation

How to find the derivative when y is not alone, horizontal and vertical tangent lines, and the power rule for fraction exponents.

Derivatives of exponential and logarithmic functions

The number e, the derivatives of exponential and logarithmic functions, growth rates, and logarithmic differentiation.

Higher derivatives

Second, third, and higher derivatives, their notation, acceleration, and patterns in repeated derivatives.

Before you start

This chapter builds on the derivative. These pages may help:

• The derivative at a point (the definition of the derivative)
• The derivative as a function (derivative notation)
• The Squeeze Theorem (the limits of sin x / x and (1 − cos x) / x)
• Integer exponents
• Rational exponents
• Composite functions
• Exponential functions
• Logarithmic functions (properties of logarithms)