1. Exponential Derivatives Brooke Smith

2. Exponential Functions and Their Derivatives  f(x)=e x f(x)=a x  f’(x)=e x f’(x)=a x (ln(a))

3. Examples of Derivatives of f(x)=e x  f(x)=4x 3 + 3x + e x  f’(x) = 12x 2 + 3 + e x  To find the derivative of the original function, the power rule and knowledge of exponential functions must be applied. Therefore, the exponent is multiplied times the coefficients, then 1 must be subtracted from the original exponent to find the derivative of 4x 3 and 3x. However, the derivative of e x is e x, so it remains the same.

4. Examples of Derivatives of f(x)=e x  f(x) = 2e x^2  f’(x) = 2x(2e x^2 )  Simplify:  f’(x) = 4xe x^2  The derivative can be found by taking the derivative of the outside (e x^2 ) multiplied by the derivative of the inside (x 2 ).

5. Examples of Derivatives of f(x)=a x  f(x)=6 x  f’(x)=6 x ((ln(6))  The derivative was found by applying the exponential function rule, shown in slide #2.

6. Examples of Derivatives of f(x)=a x  Find the derivative of f(x) = 2x 2 (4 x )  f’(x) = 4x(4 x )+((ln(4)4 x )(2x 2 )  The derivative can be found by using the product and exponential function rule. Therefore, the derivative is taken of the first term and then multiplied by the second term. Then, it is added to the derivative of the second term multiplied by the first. The derivative of the first term is found by using the power rule, while the derivative of the second term is found by using the exponential function rule.

7. Practice Problem #1  Solve for the first derivative of f(x) = 2e x

8. Practice Problem #1 Solution  f(x) = 2e x  f’(x)= 2e x  The derivative remains the same as the original function because you multiply 2 by the derivative of e x, which remains the same.

9. Practice Problem #2  Calculate f’(x) for the function of f(x) = (2e x ) 2

10. Practice Problem #2 Answer  f(x) = (2e x ) 2  f’(x) = 2(2e x )(2e x )  Simplify:  f’(x) = 8e 2x  The derivative was found by multiplying 2 by the derivative of both of the 2e x ’s. Therefore, since the derivative of 2e x remains 2e x, the solution is just multiplication of the original terms.

11. Practice Problem #3  Solve for the dy/dx for the function, f(x) = 4x 2 +2x / 8 x

12. Practice Problem #3 Answer  f(x) = 4x 2 +2x / 8 x  f’(x) = (8x+2)(8 x ) - (4x 2 +2x)((8 x )(ln(8)) / (8 x ) 2  Simplify:  f’(x) = 64x x + 16 x – ((4x 2 +2x)(8 x )(ln(8)) / (8 x ) 2  The solution was found by using the quotient and exponential function rules. So, the derivative of the first term was multiplied by the bottom term, then the derivative of the bottom was multiplied by the top and subtracted from the derivative of the top and the bottom term. Then, all of this was divided by the bottom squared.

13. Practice Problem #4  Solve for the derivative of… f(x) = 4x 2 + 6x + 2 – 2 x

14. Practice Problem #4 Solution  f(x) = 4x 2 + 6x + 2 – 2 x  f’(x) = 8x + 6 – ((2 x (ln(2)))  The solution was found by using the power rule and exponential function rule. The power rule was applied to the first three terms to find their derivatives, then the exponential function rule was applied to 2 x to complete finding the derivative of the original function.

15. Tips to Help with Exponential Functions  Remembering the product and quotient rules are vital to being able to solve exponential functions.  So, the product rule can be remembered by this shortcut…  First term × derivative of second term + derivative of first term × second term  Then, the quotient rule can be remembered by…  Bottom term × derivative of top − top × derivative of bottom, all divided by the bottom term squared

16. Exponential Derivatives  I hope that you have learned more about how to solve functions involving exponential derivatives!