find the derivative. f(x)=6√(9x² + 5) f(x)=□

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=6\sqrt{9x^{2}+5}$ as $f(x)=6(9x^{2}+5)^{\frac{1}{2}}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = f(g(x))$, then $y'=f'(g(x))\cdot g'(x)$. Let $u = 9x^{2}+5$, 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 $x$: $\frac{du}{dx}=18x$.

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

By the chain - rule $f'(x)=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 9x^{2}+5$ back into the equation: $f'(x)=3(9x^{2}+5)^{-\frac{1}{2}}\cdot18x$.

Step4: Simplify the result

$f'(x)=\frac{54x}{\sqrt{9x^{2}+5}}$.

Answer:

$\frac{54x}{\sqrt{9x^{2}+5}}$