Find the derivative of $f(x)=-10\sqrt{x^{20}+9}$ with respect to $x$
I know to take the constant out and let $u=x^{20}+9$
\begin{align}
f'=&-10 \cfrac{df}{dx}(u)^{1/2} \hspace{2cm} (1)
\end{align}
\begin{align}
f'=&-10 \cfrac{1}{2}(u)^{-1/2} \hspace{2cm} (2) \\ f'=& \cfrac{-10}{2\sqrt{u}} \hspace{3.75cm} (3) \\ f' =& \cfrac{-5}{\sqrt{x^{20}+9}} \hspace{2.8cm} (4)
\end{align}
I know this is very wrong, but I don't understand why the correct answer is $-10 \cdot \cfrac{1}{2\sqrt{x^{20}+9}}\cdot 20x^{19}$. I understand everything except the last term, $20x^{19}$. I'm aware it is the derivative of $u$, but I don't understand why we multiply by that to the numerator after having already taken the derivative of root $u$ as shown in line $2$.
can anyone explain why multiplying that term is necessary?
Best Answer
$\frac d {dx} h(g(x))=h'(g(x)) g'(x)$ by Chain Rule. You forgot $g'(x)$. [Here $h(x)=-10\sqrt x$ and $g(x)=x^{20}+9$].