Question

How do you differentiate `f(x)= (x^3+2x+1)*sqrtsinx` using the product rule?

Answer 1

`f'(x) = (3x^2 + 2) sqrt(sin x) + (x^3 + 2x + 1) * 1/(2 sqrt(sin x)) * cos x`

Explanation:

The product rule is:

If `f(x) = g(x) * h(x)`, then the derivative of `f(x)` is

`f'(x) = g'(x) * h(x) + g(x) * h'(x)`

In your case,

`g(x) = x^3 + 2x + 1`

and

`h(x) = sqrt(sin x) = (sin x)^(1/2)`

So, let's compute the derivatives of `g(x)` and `h(x)`:

`g'(x) = 3x^2 + 2`

To build the derivative of `h(x)`, you need to apply the chain rule:

`h'(x) = 1/2 (sin x) ^(-1/2 ) * cos x = 1/(2 sqrt(sin x)) * cos x`

Now, the only thing left to do is use the product rule:

`f'(x) = g'(x) * h(x) + g(x) * h'(x)`

`color(white)(xxxii)= (3x^2 + 2) sqrt(sin x) + (x^3 + 2x + 1) * 1/(2 sqrt(sin x)) * cos x`