Derivatives of Exponential Functions TS: Explicitly Assessing Information and Drawing Conclusions

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Derivatives of Exponential FunctionsTS
Explicitly Assessing Information and Drawing
Conclusions
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Objective

• To find the derivatives of natural exponential
  functions.

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Derivatives of Exponential Functions
What is the derivative?
Super Guess
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Derivatives of Exponential Functions
What is the derivative?
Super Guess
WRONG
This is incorrect.
This rule works for f (x) xN, when the base is
a variable x and the exponent is a constant N.
The power rule does not apply to exponential
functions.
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Derivatives of Exponential Functions
What is the derivative?
CORRECT!
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Derivatives of Exponential Functions

• Use the chain rule to find the derivative of the
  composition of the natural exponential function
  and another function.

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Derivatives of Exponential Functions

• The derivative of an exponential function of the
  form f (x) eu is the product of the function
  and the derivative of its exponent.

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Derivatives of Exponential Functions

• Differentiate

Derivative of 5x is 5
Derivative of eu is eu
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Derivatives of Exponential Functions

• Differentiate

Derivative of x3 is 3x2
Derivative of eu is eu
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Derivatives of Exponential Functions

• Differentiate

Derivative of 3x2 is 6x
Derivative of eu is eu
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Derivatives of Exponential Functions

• Differentiate

Derivative of x-3 is -3x-4
Derivative of eu is eu
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Derivatives of Exponential Functions

• Differentiate

Product Rule
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Derivatives of Exponential Functions

• Differentiate

Quotient Rule
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Derivatives of Exponential Functions

• Differentiate

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Derivatives of Exponential Functions

• Differentiate

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Derivatives of Exponential Functions

• Differentiate

Product Rule
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Derivatives of Exponential Functions

• Differentiate

Quotient Rule
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Conclusion

• The derivative of the natural exponential
  function is itself.
• The derivative of an exponential function of the
  form f (x) eu is the product of the function
  and the derivative of its exponent.