Question

find the derivative of the function.

$f(x)=sin^{-1}(6x)$

$f(x)=$

Explanation:

Step1: Recall 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 $u = 6x$, so $y=\sin^{-1}(u)$.

Step2: Find the derivative of $y$ with respect to $u$

The derivative of $y = \sin^{-1}(u)$ with respect to $u$ is $\frac{dy}{du}=\frac{1}{\sqrt{1 - u^{2}}}$.

Step3: Find the derivative of $u$ with respect to $x$

Since $u = 6x$, then $\frac{du}{dx}=6$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $\frac{dy}{du}=\frac{1}{\sqrt{1 - u^{2}}}$ and $\frac{du}{dx}=6$ into the chain - rule formula. Replace $u$ with $6x$, we get $\frac{dy}{dx}=\frac{6}{\sqrt{1-(6x)^{2}}}$.

Answer:

$\frac{6}{\sqrt{1 - 36x^{2}}}$