Derivatives

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Derivatives

• Chapter 3

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Chain Rule

• 3.6

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Concepts in 3.6

• Derivative of a Composite Function
• Outside-Inside Rule
• Repeated Use of the Chain Rule
• Slopes of Parametrized Curves
• Power Chain Rule
• and why
• The chain rule is the most widely used
  differentiation
• rule in mathematics.

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Derivative of a Composite Function
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Derivative of a Composite Function
y 3u u 3x5 y 9x15
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Derivative of a Composite Function
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Derivative of a Composite Function
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Derivatives of Composite Functions
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Rule 8 The Chain Rule
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Chain Rule
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Outside-Inside Rule

• Find the derivative of the outside function
  leaving the inside function where it is and the
  same. Then multiply that by the derivative of
  the inside.

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Example Derivatives of Composite Functions
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Example Derivatives of Composite Functions
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Example Outside-Inside Rule

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Example Repeated Use of the Chain Rule

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Example Repeated Use of the Chain Rule

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Slopes of Parametrized Curves

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Slopes of Parametrized Curves

• dy/dt (dy/dx)(dx/dt)
• Assuming each derivative exists and by the chain
  rule.

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Finding dy/dx Parametrically
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Example--Finding dy/dx Parametrically
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Example--Finding dy/dx Parametrically
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Power Chain Rule
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Radian vs. Degrees

• The formulas for the derivatives of sin x and cos
  x were obtained under the assumption that x is
  measured in radians and not degrees. Use the
  chain rule to see why