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

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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  1. Chapter 4Techniques of DifferentiationSections 4.1, 4.2, and 4.3

  2. Techniques of Differentiation • The Product and Quotient Rules • The Chain Rule • Derivatives of Logarithmic and Exponential asFunctions

  3. Available Rules for Derivatives

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

  5. Derivative of first Derivative of Second The Product Rule - Example

  6. Derivative of denominator Derivative of numerator The Quotient Rule - Example

  7. 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.

  8. 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.

  9. 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.

  10. Generalized Power Rule Example:

  11. Generalized Power Rule Example:

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

  13. Sub in for u Chain Rule Example

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

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

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

  17. Examples

  18. Logarithms of Absolute Values

  19. Examples

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

  21. Examples Find the derivative of Find the derivative of

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

  23. Exponential Functions Find the derivative of

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