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Techniques of Differentiation

Techniques of Differentiation. The Product and Quotient Rules The Chain Rule Derivatives of Logarithmic and Exponential Functions Implicit Differentiation. The Product Rule. The Quotient Rule. The Product Rule. Ex. Derivative of Second. Derivative of first. The Quotient Rule.

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Techniques of Differentiation

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  1. Techniques of Differentiation • The Product and Quotient Rules • The Chain Rule • Derivatives of Logarithmic and Exponential Functions • Implicit Differentiation

  2. The Product Rule The Quotient Rule

  3. The Product Rule Ex. Derivative of Second Derivative of first

  4. The Quotient Rule Ex. Derivative of denominator Derivative of numerator

  5. Compute the Derivative Ex. = –10

  6. 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.

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

  8. 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.

  9. Generalized Power Rule Ex.

  10. The Chain Rule Ex.

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

  12. Chain Rule Example Ex. Sub in for u

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

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

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

  16. Differentiation of Logarithmic Functions Ex.

  17. Derivative of Logarithms of Absolute Values

  18. Derivative of Logarithms of Absolute Values Ex. Ex.

  19. Differentiation of Exponential Functions Derivative of ex: Generalized Rule for eu: If u is a differentiable function, then

  20. Derivatives of Exponential Functions Ex. Find the derivative of Ex. Find the derivative of

  21. Differentiation of Exponential Functions Derivative of bx: Generalized Rule for bu: If u is a differentiable function, then

  22. Derivatives of Exponential Functions Ex. Find the derivative of

  23. Implicit Differentiation y is explicitly a function of x. y is implicitly a function of x.

  24. Implicit Differentiation (cont.) To differentiate the implicit case we use the chain rule where y is a function of x: Solve for

  25. Tangent Line to Implicit Curve Ex. Find the equation of the tangent line to the curve at the point (2, 1).

  26. Logarithmic Differentiation Ex. Use logarithmic differentiation to find the derivative of Apply ln Properties of ln Differentiate Solve

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