Question

How do you differentiate `f(x) = xe^(-2x^2)` using the product rule?

Answer 1

`f'(x)=(1-4x^2)/(e^(2x^2))`

Explanation:

We have two terms: `x` and `e^(-2x^2)`. The product rule states that the derivative of the product of the two will equal the following:
`f'(x)=color(red)(d/(dx)[x]*e^(-2x^2))+color(blue)(x*d/(dx)[e^(-2x^2)])`

Let's take a moment to calculate both of the derivatives inside of our product rule statement.
`color(red)(d/(dx)[x]=1)`
The next derivative will require the chain rule. Also, remember that the derivative of `e^x` is `e^x`.
`color(blue)(d/(dx)[e^(-2x^2)]=d/(dx)[-2x^2]*e^(-2x^2)=-4x(e^(-2x^2))`

Now that we know what both our derivatives are equal to, let's plug them back into our equation.

`f'(x)=color(red)(1*e^(-2x^2))+color(blue)(x*-4x(e^(-2x^2)))`
`f'(x)=e^(-2x^2)-4x^(2)(e^(-2x^2))`
`f'(x)=e^(-2x^2)(1-4x^2)`
`color(green)(f'(x)=(1-4x^2)/(e^(2x^2)))`