1. Derivatives of Exponential Functions TS: 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 form f (x) = eu is the product of the function and the derivative of its exponent.

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

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

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

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

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 form f (x) = eu is the product of the function and the derivative of its exponent.