Chapter 4 Techniques of Differentiation Sections 4.1, 4.2, and 4.3

Techniques of Differentiation  The Product and Quotient Rules  The Chain Rule  Derivatives of Logarithmic and Exponential as Functions

Available Rules for Derivatives

Two More Rules The product rule The quotient rule If f (x) and g (x) are differentiable functions, then we have

The Product Rule - Example Derivative of first Derivative of Second

Derivative of numerator Derivative of denominator The Quotient Rule - Example

Calculation Thought Experiment Given an expression, consider the steps you would use in computing its value. If the last operation is multiplication, treat the expression as a product; if the last operation is division, treat the expression as a quotient; and so on.

Example: To compute a value, first you would evaluate the parentheses then multiply the results, so this can be treated as a product. To compute a value, the last operation would be to subtract, so this can be treated as a difference. Calculation Thought Experiment Example:

The Chain Rule The derivative of a f (quantity) is the derivative of f evaluated at the quantity, times the derivative of the quantity. If f is a differentiable function of u and u is a differentiable function of x, then the composite f (u) is a differentiable function of x, and

Generalized Power Rule Example:

Generalized Power Rule Example:

Chain Rule in Differential Notation If y is a differentiable function of u and u is a differentiable function of x, then

Chain Rule Example Sub in for u

Logarithmic Functions Generalized Rule for Natural Logarithm Functions Derivative of the Natural Logarithm If u is a differentiable function, then

Find the derivative of Find an equation of the tangent line to the graph of Slope: Equation: Examples

Generalized Rule for Logarithm Functions. Derivative of a Logarithmic Function. If u is a differentiable function, then More Logarithmic Functions

Examples

Logarithms of Absolute Values

Examples

Exponential Functions Generalized Rule for the natural exponential function. Derivative of the natural exponential function. If u is a differentiable function, then

Find the derivative of Examples

Generalized Rule for general exponential functions. Derivative of general exponential functions. If u is a differentiable function, then Exponential Functions

Find the derivative of Exponential Functions