The derivative of $f(x^2)$

Question

I wanted to ask something. Let there be a function $f(x^2)$, then the first derivative is $2 x f'(x^2)$ or only $f'(x^2)$ ? Its just something I want to be sure about cause it's some kinda confusing me. Thanks in advance

Answer 1

Let $u=x^{2}$. Then, $f(x^{2})=f(u)$. You want to differentiate $f$ with respect to $x$. By the chain rule $$ \frac{df}{dx}=\frac{df}{du}\cdot\frac{du}{dx}=f'(u)\cdot(2x)=2x\cdot f'(x^{2}). $$

Answer 2

You should consider the function $f(x^2)$ as a function of $x$, so you should look at it as $h(x)=f(x^2)$, which you can see as $h(x)=f(g(x))=f\circ g(x)$ where $g(x)=x^2$. Thus $h'(x)=(f(x^2))'=g'(x)f'(g(x))=2xf'(x^2)$