find the derivative.\nf(x)=6\\sqrt{6x^{2}+5}\nf(x)=□

Explanation:

Step1: Rewrite the function

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

Step2: Apply the constant - multiple rule

The constant - multiple rule states that if $y = cf(x)$, then $y'=cf'(x)$. Here $c = 6$ and $y=(6x^{2}+5)^{\frac{1}{2}}$. So $f'(x)=6\times\frac{d}{dx}(6x^{2}+5)^{\frac{1}{2}}$.

Step3: Apply the chain rule

Let $u = 6x^{2}+5$, then $y = u^{\frac{1}{2}}$. The chain rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, $\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}$ and $\frac{du}{dx}=12x$.

Step4: Substitute and simplify

Substitute $u = 6x^{2}+5$ back into the chain - rule result. $\frac{dy}{du}\cdot\frac{du}{dx}=\frac{1}{2}(6x^{2}+5)^{-\frac{1}{2}}\cdot12x$. Then $f'(x)=6\times\frac{1}{2}(6x^{2}+5)^{-\frac{1}{2}}\cdot12x$. Simplify the expression: $f'(x)=3\times12x(6x^{2}+5)^{-\frac{1}{2}}=\frac{36x}{\sqrt{6x^{2}+5}}$.

Answer:

$\frac{36x}{\sqrt{6x^{2}+5}}$