Question

How do you differentiate `f(x)=(1/x^3)sinx` using the product rule?

Answer 1

`-3x^-4sinx+x^-3cosx`

Explanation:

Let's start by rewriting `f(x)` as `x^(-3)sinx`. This allows us to differentiate it more easily.

We understand that if we have two functions, `g(x)` and `h(x)` being multiplied, the derivative is equal to

`g'(x)h(x)+g(x)h'(x)`

`f(x)` is essentially composed of the following:

• `color(blue)(g(x)=x^-3)`
• `color(red)(h(x)=sinx)`

We can use the power rule and our knowledge of trig derivatives to figure out that

• `color(purple)(g'(x)=-3x^-4)`
• `color(lime)(h'(x)=cosx)`

Now, we just plug this business into our expression for the product rule. We get

`color(purple)(-3x^-4)color(red)((sinx))+color(blue)(x^-3)color(lime)((cosx))`

And I can write this with neutral colors to get

`bar(ul(|color(white)(2/2)-3x^-4sinx+x^-3cosxcolor(white)(2/2)|))`

Hope this helps!