Question

find dy.
y = sin (7x²)
dy = □ dx

Explanation:

Step1: Apply the chain rule

The chain rule states that if \(y = f(g(x))\), then \(y^\prime=f^\prime(g(x))\cdot g^\prime(x)\). Let \(u = 7x^{2}\), so \(y=\sin(u)\). The derivative of \(\sin(u)\) with respect to \(u\) is \(\cos(u)\), and the derivative of \(u = 7x^{2}\) with respect to \(x\) is \(u^\prime=14x\).

Step2: Substitute back

Substitute \(u = 7x^{2}\) into the derivative. We get \(y^\prime=\cos(7x^{2})\cdot14x\).

Answer:

\(14x\cos(7x^{2})\)