Calculus: When do we use this rule? $\left[u\left(x\right)^n\right]'=n\:u\left(x\right)^{\left(n-1\right)}\:\cdot \:u'\left(x\right)$

Question

If we want to solve the following question: $$\frac{d}{dx}\left(\frac{x}{\sqrt{2x-1}}\right)$$ Then we will have to use this following rule (the power rule), which seems strange to me since using the power rule for other equations doesn't require the same step to be performed when using this equation, which is: $$\left[u\left(x\right)^n\right]'=n \:u\left(x\right)^{\left(n-1\right)}\:\cdot \:u'\left(x\right)$$

So to summarize my question, when do we use this rule and why do we do so? (Couldn't fit the question into the title bar because it was long)

Answer 1

Ok, I see the problem. Your problem is with the $'$ notation. Let's use another:

$$\frac{du^n(x)}{dx}=n u^{n-1}(x)\frac{du(x)}{dx}$$

Now if instead you were taking an $u$ derivative:

$$\frac{du^n}{du}=n u^{n-1}\frac{du}{du}=n u^{n-1}$$

It all depends on the variable you are differentiating with respect to.

Answer 2

In general, $$ f(g(x))'=f'(g(x))g'(x) $$ by the chain rule. You are asking about the case when $f(x)=x^n$.