what does $f '(x^2)$ mean?

Question

enter image description herewhat does $f '(x^2)$ mean? does it mean calculating $f '(x)$ and then putting x=x² or first putting $x=x^2$ in $f(x)$ and then differentiating it?

Let $f(x)=x^3$ and $f'(x)=g(x)$ So $g(x)=3x^2$

$g(x^2)= 3x^4$

$f'(x^2)=3x^4$ how this is incorrect?

Answer 1

You are correct!

DON'T substitute $x=x^2$ in $f(x)$. $$f(x)=x^3$$
Take first derivative: $$f'(x)=3x^2$$ Replace $x$ by $x^2$ $$f'(x^2)=3(x^2)^2=3x^4$$ This is how you find value of $f'(x)$ for ANY $x$; just replace $x$ by that.
Who told you $3x^4$ is incorrect?

Answer 2

$f'(x^2)$ means

1. find the derivative of $f(x)$. $f'(x)=g(x)$
2. evaluate the derivate at point $x^2$. $f'(x^2)=g(x^2)$

Answer 3

The derivative of $h(x)=f(x^2)$ should be denoted $h'(x)=\bigl(f(x^2)\bigr)'$, whereas $f'(x^2)$ is the derivative of $f$, evaluated at $x^2$. So your answer is correct.