I came across this problem that's incredibly confusing and I have no clue if my process was correct.
So the question went as follows: "Let $f$ be an increasing function with $f(0) = 3$. The derivative of $f$ is given by $f'(x)=\cos(\pi x)+x^4+6$."
So I was then asked the following: "let $g$ be the function defined by $g(x) = f(\sqrt{2x^2+7}))$. Find $g'(3)$."
So I thought the correct way to go about this way by substituting the $\sqrt{2x^2+7}$ for $x$ in the original equation for $f(x)$, but I'm not all that sure. Any help would be appreciated!
Best Answer
You can compute the antiderivative of $f$. Also, you even know the corresponding constant in the antiderivative, you can find it using the condition $f(0) = 3$. After that it's not difficult to write the formula for $g(x)$, take its derivative and calculate $g^{'}(3)$.
One more to solve it is to use chain rule, say that $t = \sqrt{2 x^2 + 7}$, then $g(x) = f(t(x))$ and $g^{'}(x) = f^{'}(t) \, t^{'}(x)$