Find the derivative of the function for the given value of x

Question

The function is $\left[{\sqrt[3]{ax^2}+\sqrt[3]{a^2x}}\right]$ and ${x=a}$

I'm confused on what to do first, should I substitute ${x}$ for ${a}$ and then get the derivative or is it the other way around?

Answer 1

$f(x) = \sqrt[3]{ax^2}+\sqrt[3]{a^2x} = a^{1/3}x^{2/3} + a^{2/3}x^{1/3}$

$\cfrac{df(x)}{dx} = \cfrac{2a^{1/3}}{3x^{1/3}} + \cfrac{a^{2/3}}{3x^{2/3}} = \cfrac{2\sqrt[3]{a}}{3\sqrt[3]{x}} + \cfrac{\sqrt[3]{a^2}}{3\sqrt[3]{x^2}}$

If you substitute $x = a$ in the derivative, the roots will cancel out and you will get $\frac{2}{3} + \frac{1}{3} = 1$.