Question

if ( y = sin^{-1}(5x) ), then ( \frac{dy}{dx} = )

Explanation:

Step1: Apply the chain rule

Let \(u = 5x\), then \(y=\sin^{-1}(u)\). The derivative of \(\sin^{-1}(u)\) with respect to \(u\) is \(\frac{1}{\sqrt{1 - u^{2}}}\), and the derivative of \(u = 5x\) with respect to \(x\) is \(5\).

Step2: Combine using the chain rule formula \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\)

\(\frac{dy}{dx}=\frac{1}{\sqrt{1-(5x)^{2}}}\cdot5\)

Answer:

\(\frac{5}{\sqrt{1 - 25x^{2}}}\)