Chain rule for differentiation

Question

I've been given this problem:

$y= \sqrt{7+6x^3}$

Using the chain rule am I right in suggesting that

$$u = 7+6x^3$$ $$y = \sqrt{u}$$

Answer 1

Yes, there are two ways to see it: $$\frac{{\rm d}y}{{\rm d}x} = \frac{{\rm d}y}{{\rm d}u} \frac{{\rm d}u}{{\rm d}x} = \frac{1}{2\sqrt{u}}18x^2 = \frac{9x^2}{\sqrt{7+6x^3}},$$or calling $f(x) = \sqrt{x}$ and $g(x) = 7+6x^3$, we have that $y(x) = f(g(x))$, so: $$y'(x) = f'(g(x))g'(x) = \frac{1}{2\sqrt{g(x)}}g'(x) = \frac{1}{2\sqrt{7+6x^3}}18x^2 = \frac{9x^2}{\sqrt{7+6x^3}}. $$