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Objective

Learn to find derivatives of natural exponential functions accurately using chain rule, product rule, and more. Understand the uniqueness and rules of differentiation for exponential functions.

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Objective

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  1. Derivatives of Exponential FunctionsTS: Explicitly Assessing Information and Drawing Conclusions

  2. Objective • To find the derivatives of natural exponential functions.

  3. Derivatives of Exponential Functions What is the derivative? Super Guess:

  4. 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.

  5. Derivatives of Exponential Functions What is the derivative? CORRECT!

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

  7. Derivatives of Exponential Functions • The derivative of an exponential function of the formf (x) = euis the product of the function and the derivative of its exponent.

  8. Derivatives of Exponential Functions • Differentiate: Derivative of 5x is 5 Derivative of eu is eu

  9. Derivatives of Exponential Functions • Differentiate: Derivative of x3 is 3x2 Derivative of eu is eu

  10. Derivatives of Exponential Functions • Differentiate: Derivative of 3x2 is 6x Derivative of eu is eu

  11. Derivatives of Exponential Functions • Differentiate: Derivative of x-3 is -3x-4 Derivative of eu is eu

  12. Derivatives of Exponential Functions • Differentiate: Product Rule

  13. Derivatives of Exponential Functions • Differentiate: Quotient Rule

  14. Derivatives of Exponential Functions • Differentiate:

  15. Derivatives of Exponential Functions • Differentiate:

  16. Derivatives of Exponential Functions • Differentiate: Product Rule

  17. Derivatives of Exponential Functions • Differentiate: Quotient Rule

  18. Conclusion • The derivative of the natural exponential function is itself. • The derivative of an exponential function of the formf (x) = euis the product of the function and the derivative of its exponent.

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