Question

find the derivative.
f(x)=4\sqrt{6x^{2}+7}
f(x)=□

Explanation:

Step1: Rewrite the function

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

Step2: Apply the chain - rule

The chain - rule states that if $y = f(g(x))$, then $y'=f'(g(x))\cdot g'(x)$. Let $u = 6x^{2}+7$, so $y = 4u^{\frac{1}{2}}$. First, find the derivative of $y$ with respect to $u$: $\frac{dy}{du}=4\times\frac{1}{2}u^{-\frac{1}{2}} = 2u^{-\frac{1}{2}}$. Then find the derivative of $u$ with respect to $x$: $\frac{du}{dx}=12x$.

Step3: Calculate the derivative of $f(x)$

By the chain - rule $f'(x)=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 6x^{2}+7$ back into the formula: $f'(x)=2(6x^{2}+7)^{-\frac{1}{2}}\times12x=\frac{24x}{\sqrt{6x^{2}+7}}$.

Answer:

$\frac{24x}{\sqrt{6x^{2}+7}}$