Question

How do you differentiate the equation?

`e^x(x^3 + 2x) = f(x)`

Answer 1

`f'(x) = (e^x)(x^3+2x + 3x^2 + 2)`

Explanation:

Step 1) You know that you must use the product rule because you are trying to find the derivative of a product.

Step 2) So, you do the derivative of the first (`e^x`) times the second `(x^3+2x)`, plus the first (`e^x`) times the derivative of the second `(x^3+2x)`.

`f'(x) = d/dx(e^x)(x^3+2x) + (e^x)d/dx(x^3+2x)`

Step 3) Now you must find simplify and find each derivative.

Step 4) The derivative of `e^x` is always `e^x`, and the derivative of `(x^3+2x)` is `(3x^2 + 2)`. Insert those into the equation.

`f'(x) = (e^x)(x^3+2x) + (e^x)(3x^2 + 2)`

Step 5) Pull out the `e^x`, giving you your final answer.

`f'(x) = (e^x)(x^3+2x + 3x^2 + 2)`

Answer 2

`d/dx(e^x(x^3+2x))=e^x(x^3+2x)+e^x(3x^2+2)`

Explanation:

Apply the product rule: `d/dx(f*g)=(f'*g)+(f*g')`

Let `f=e^x` and `g=x^3+2x`

`d/dx(e^x(x^3+2x))=d/dx(e^x)(x^3+2x)+(e^x)(d/dx(x^3+2x))`

`=e^x(x^3+2x)+e^x(3x^2+2)`