Question

What is the derivative of `g(x)=x^3 cos x`?

Answer 1

`g'(x)= 3x^2cosx-x^3sinx`

Explanation:

Since `g(x)` is the product of two terms, we can use the Product Rule to find the derivative.

We essentially have `g(x)=f(x)*h(x)`, where

`color(purple)(f(x)=x^3)` and

`color(green)(h(x)=cosx)`

Thus, the Product Rule states that the derivative is equal to:

`f'(x)h(x)+f(x)h'(x)`

To differentiate `f(x)`, we can use the Power Rule, where the exponent becomes the coefficient, and we decrement the power. Thus,

`color(blue)(f'(x)=3x^2)`

And from our knowledge of derivatives of trig functions

`color(red)(h'(x)=-sinx)`

We can now plug these values into the product rule expression to get

`color(blue)(3x^2)color(green)((cosx))+color(purple)(x^3)color(red)((-sinx))`

We can rewrite this as

`3x^2(cosx)-x^3(sinx)`

Thus, `g'(x)= 3x^2cosx-x^3sinx`

If the Power or Product Rules seem foreign to you, I encourage you to Google them or go to Khan Academy to understand them more.

Hope this helps!