Question

How do you differentiate `g(x) = (x/2)e^(2x)` using the product rule?

Answer 1

`(dg)/(dx)=e^(2x)/2+xe^(2x)`

Explanation:

Product rule states if `g(x)=f(x)h(x)`

then `(dg)/(dx)=(df)/(dx)xxh(x)+(dh)/(dx)xxf(x)`

Hence as `g(x)=(x/2)e^(2x)`

`(dg)/(dx)=1/2xxe^(2x)+2xxe^(2x)xxx/2`

Here we have also used the concept of function of a function and used chain rule for it. As we have differentiated `e^(2x)` w.r.t. `2x`, we must multiply by differential of `2x` w.r.t `x` i.e. by `2`.

So `(dg)/(dx)=e^(2x)/2+xe^(2x)`