Question

(a) Find f'(x) if

f(x) = ((2x ^ 2 - 1)(x ^ 2 + 3sqrt(x)))/(x ^ 3 + 2sqrt(x))

Answer 1

That is very, very difficult to solve and write on this site.

In google type:

online derivative calculator with steps

When you see list of results click on:

Online Derivative Calculator with Steps - eMathHelp

When page be open in rectangle Function paste:

((2x ^ 2 - 1)(x ^ 2 + 3sqrt(x)))/(x ^ 3 + 2sqrt(x))

then click CALCULATE

you will see solution step-by-step

Answer 2

f(x) = ((2x^2 - 1)(x^2 + 3√x))/(x^3 + 2√x)
consider this as f(x) = u/v where u and v are functions of x. Then, using the quotient rule,
f'(x) = (u'v - uv')/v^2
Now, u' can be found using the product rule, so we have
f'(x) = (((4x)(x^2 + 3√x))+(3x^2-1)(2x + 3/(2√x)))(x^2 + 3√x) - ((2x^2 - 1)(x^2 + 3√x))(3x^2 + 1/√x)) / (x^3 + 2√x)^2

Now simplify that! You should get something like

x[4x^5+2x^3+48 + √x (28x^2+15x-6)]
-------------------------------------------------
2(x^3 + 2√x)^2