find the derivative of the function.\n$f(t)=sqrt{6t^{2}-t}$\n$f(t)=$

Explanation:

Step1: Rewrite the function

Rewrite $f(t)=\sqrt{6t^{2}-t}=(6t^{2}-t)^{\frac{1}{2}}$.

Step2: Apply the chain - rule

The chain - rule states that if $y = u^{\frac{1}{2}}$ and $u = 6t^{2}-t$, then $\frac{dy}{dt}=\frac{dy}{du}\cdot\frac{du}{dt}$. First, find $\frac{dy}{du}$: $\frac{d}{du}(u^{\frac{1}{2}})=\frac{1}{2}u^{-\frac{1}{2}}$. Second, find $\frac{du}{dt}$: $\frac{d}{dt}(6t^{2}-t)=12t - 1$.

Step3: Calculate the derivative

$\frac{dy}{dt}=\frac{1}{2}(6t^{2}-t)^{-\frac{1}{2}}\cdot(12t - 1)=\frac{12t - 1}{2\sqrt{6t^{2}-t}}$.

Answer:

$\frac{12t - 1}{2\sqrt{6t^{2}-t}}$