chapter 4 techniques of differentiation sections 4 1 4 2 and 4 3 n.
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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 .

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slide2

Techniques of Differentiation

  • The Product and Quotient Rules
  • The Chain Rule
  • Derivatives of Logarithmic and Exponential asFunctions
slide4

Two More Rules

If f(x) and g(x) are differentiable functions, then we have

The product rule

The quotient rule

slide5

Derivative of first

Derivative of Second

The Product Rule - Example

slide6

Derivative of denominator

Derivative of numerator

The Quotient Rule - Example

slide7

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.

slide8

Calculation Thought Experiment

Example:

To compute a value, first you would evaluate the parentheses then multiply the results, so this can be treated as a product.

Example:

To compute a value, the last operation would be to subtract, so this can be treated as a difference.

slide9

The Chain Rule

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

The derivative of a f (quantity) is the derivative of f evaluated at the quantity, times the derivative of the quantity.

slide12

Chain Rule in Differential Notation

If y is a differentiable function of u and u is a differentiable function of x, then

slide13

Sub in for u

Chain Rule Example

logarithmic functions
Logarithmic Functions

Derivative of the Natural Logarithm

Generalized Rule for Natural Logarithm Functions

If u is a differentiable function, then

examples
Examples

Find the derivative of

Find an equation of the tangent line to the graph of

Equation:

Slope:

more logarithmic functions
More Logarithmic Functions

Derivative of a Logarithmic Function.

Generalized Rule for Logarithm Functions.

If u is a differentiable function, then

exponential functions
Exponential Functions

Derivative of the natural exponential function.

Generalized Rule for the natural exponential function.

If u is a differentiable function, then

examples1
Examples

Find the derivative of

Find the derivative of

exponential functions1
Exponential Functions

Derivative of general exponential functions.

Generalized Rule for general exponential functions.

If u is a differentiable function, then

exponential functions2
Exponential Functions

Find the derivative of