Question

find the derivative.
f(x)=6\sqrt{4x^{2}+9}
f(x)=□

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=6\sqrt{4x^{2}+9}$ as $f(x) = 6(4x^{2}+9)^{\frac{1}{2}}$.

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 $u = 4x^{2}+9$, so $y = 6u^{\frac{1}{2}}$. First, find the derivative of $y$ with respect to $u$: $\frac{dy}{du}=6\times\frac{1}{2}u^{-\frac{1}{2}} = 3u^{-\frac{1}{2}}$. Then find the derivative of $u$ with respect to $x$: $\frac{du}{dx}=8x$.

Step3: Calculate the derivative of $y$ with respect to $x$

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 4x^{2}+9$ back into the equation: $\frac{dy}{dx}=3(4x^{2}+9)^{-\frac{1}{2}}\cdot8x$.

Step4: Simplify the result

$\frac{dy}{dx}=\frac{24x}{\sqrt{4x^{2}+9}}$.

Answer:

$\frac{24x}{\sqrt{4x^{2}+9}}$