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

↳Redirected from "Suppose that I don't have a formula for #g(x)# but I know that #g(1) = 3# and #g'(x) = sqrt(x^2+15)# for all x. How do I use a linear approximation to estimate #g(0.9)# and #g(1.1)#?"
1 Answer
Lotusbluete
Dec 15, 2015

#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#

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