# Derivative of polynomial root function

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?

• Did you use the chain rule properly? Dec 25 '19 at 5:21
• $$\dfrac{d(f(x))}{dx} \ne \dfrac{df(u)}{dx}$$ Dec 25 '19 at 5:29

$$\frac d {dx} h(g(x))=h'(g(x)) g'(x)$$ by Chain Rule. You forgot $$g'(x)$$. [Here $$h(x)=-10\sqrt x$$ and $$g(x)=x^{20}+9$$].
Let, $$u = x^{20}+9$$
$$f'(x)=-10 \cfrac{d(u)^{\frac 12}}{dx}$$ $$f'(x)=-10 \cfrac{d(u)^{\frac 12}}{du} \cfrac{du}{dx}$$ $$f'(x)=-10 \cfrac{u^{\frac {-1}{2}}}{2} \cfrac{du}{dx}$$ And we know that $$\cfrac {du}{dx} = 20x^{19}$$ Hence, $$f'(x)= \cfrac {-5}{\sqrt {x^{20}+9}} 20x^{19}$$ Therefore, $$f'(x)= -10 \cfrac{1}{2 \sqrt {x^{20}+9}} 20x^{19}$$