Question

homework 3.6 the chain rule
score: 20/100 answered: 2/10
question 3
textbook videos +
use the chain rule to find the derivative of
f(x)=4sqrt{2x^{10}+7x^{6}}
type your answer without fractional or negative exponents.
f(x)=
question help: video

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=4\sqrt{2x^{10}+7x^{6}}$ as $f(x) = 4(2x^{10}+7x^{6})^{\frac{1}{2}}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = f(g(x))$, then $y^\prime=f^\prime(g(x))\cdot g^\prime(x)$. Let $u = 2x^{10}+7x^{6}$, so $y = 4u^{\frac{1}{2}}$. First, find $\frac{dy}{du}$ and $\frac{du}{dx}$.
For $y = 4u^{\frac{1}{2}}$, $\frac{dy}{du}=4\times\frac{1}{2}u^{-\frac{1}{2}} = 2u^{-\frac{1}{2}}$.
For $u = 2x^{10}+7x^{6}$, $\frac{du}{dx}=2\times10x^{9}+7\times6x^{5}=20x^{9}+42x^{5}$.

Step3: Calculate $f^\prime(x)$

By the chain - rule $f^\prime(x)=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 2x^{10}+7x^{6}$ back into $\frac{dy}{du}$ and multiply by $\frac{du}{dx}$:
$f^\prime(x)=2(2x^{10}+7x^{6})^{-\frac{1}{2}}\cdot(20x^{9}+42x^{5})=\frac{2(20x^{9}+42x^{5})}{\sqrt{2x^{10}+7x^{6}}}=\frac{40x^{9}+84x^{5}}{\sqrt{2x^{10}+7x^{6}}}$.
To get rid of the square - root in the denominator, multiply the numerator and denominator by $\sqrt{2x^{10}+7x^{6}}$:
$f^\prime(x)=\frac{(40x^{9}+84x^{5})\sqrt{2x^{10}+7x^{6}}}{2x^{10}+7x^{6}}$.

Answer:

$\frac{(40x^{9}+84x^{5})\sqrt{2x^{10}+7x^{6}}}{2x^{10}+7x^{6}}$