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

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# Chapter 4 Techniques of Differentiation Sections 4.1, 4.2, and 4.3 - PowerPoint PPT Presentation

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

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Techniques of Differentiation

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

Two More Rules

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

The product rule

The quotient rule

Derivative of first

Derivative of Second

The Product Rule - Example

Derivative of denominator

Derivative of numerator

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.

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.

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.

Chain Rule in Differential Notation

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

Sub in for u

Chain Rule Example

Logarithmic Functions

Derivative of the Natural Logarithm

Generalized Rule for Natural Logarithm Functions

If u is a differentiable function, then

Examples

Find the derivative of

Find an equation of the tangent line to the graph of

Equation:

Slope:

More Logarithmic Functions

Derivative of a Logarithmic Function.

Generalized Rule for Logarithm Functions.

If u is a differentiable function, then

Exponential Functions

Derivative of the natural exponential function.

Generalized Rule for the natural exponential function.

If u is a differentiable function, then

Examples

Find the derivative of

Find the derivative of

Exponential Functions

Derivative of general exponential functions.

Generalized Rule for general exponential functions.

If u is a differentiable function, then

Exponential Functions

Find the derivative of