Question

Question #99f2e

Answer 1

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

Explanation:

We will use the following:

• The product rule `d/dx f(x)g(x) = f(x)g'(x)+g(x)f'(x)`

• The sum rule `d/dx f(x)+g(x) = f'(x)+g'(x)`

• `d/dxx^n = nx^(n-1)`

• `d/dx cf(x) = cf'(x)`

• `d/dxe^x = e^x`

With those:

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

`=(x^3+2x)e^x+e^x((d/dxx^3)+(d/dx2x))`

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

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

Answer 2

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

Explanation:

There's a fun little trick for differentiating functions that are multiplied by `e^x` that we can derive through the product rule.

If we have some function:

`f(x)=g(x)e^x`

Then, through the product rule:

`f'(x)=g'(x)e^x+g(x)e^x=e^x(g(x)+g'(x))`

In other words, if a function is multiplied by `e^x`, its derivative is `e^x` times the inner function plus that inner function's own derivative.

Since said inner function here is `x^3+2x`, and its derivative through the power rule is `3x^2+2`, we see that:

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