Question

How do you differentiate `f(x) = (x/4)(e^(-x/4))` using the product rule?

Answer 1

The product rule is:

For `f(x) = u(x) * v(x)`, the derivative can be built as follows:

`f'(x) = u'(x) * v(x) + u(x) * v'(x)`

Let's say that in your example, `u(x) = x/4` and `v(x) = e^(-x/4)`.

Build the derivatives for `u(x)` and `v(x)` first:

`u(x) = x/4 color(white)(xxxx) => u'(x) = 1/4`
`v(x) = e^(-x/4) color(white)(xxx) => v'(x) = -1/4 e^(-x/4)`

So, let's compute the derivative of `f(x)`:

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

`color(white)(xxxx) = (1/4 - x/16) e^(-x/4)`