Question

find dy.
y = sin(15x²)
dy = □ dx

Explanation:

Step1: Apply the chain rule

The chain rule states that if \(y = f(g(x))\), then \(y^\prime=f^\prime(g(x))\cdot g^\prime(x)\). Here \(y = \sin(u)\) with \(u = 15x^{2}\). The derivative of \(\sin(u)\) with respect to \(u\) is \(\cos(u)\), and the derivative of \(u = 15x^{2}\) with respect to \(x\) is \(u^\prime=30x\).

Step2: Substitute back \(u = 15x^{2}\)

By the chain rule, \(\frac{dy}{dx}=\cos(15x^{2})\cdot30x\). Then, since \(dy=\frac{dy}{dx}dx\), we have \(dy = 30x\cos(15x^{2})dx\).

Answer:

\(30x\cos(15x^{2})\)