Question

1. $f(x)=6cos x^{2}$

$\therefore 12xsin (x^{2})$
e. $-2xsin (x^{2})$
l. $-6xsin (x^{2})$
m. $-12xsin (x^{2})$

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 \(u = x^{2}\), so \(f(x)=6\cos(u)\). The derivative of \(\cos(u)\) with respect to \(u\) is \(-\sin(u)\), and the derivative of \(u = x^{2}\) with respect to \(x\) is \(2x\).

Step2: Calculate the derivative

First, \(f^\prime(u)=- 6\sin(u)\) (derivative of \(6\cos(u)\) with respect to \(u\)). Then, by the chain rule \(f^\prime(x)=f^\prime(u)\cdot\frac{du}{dx}\). Substituting \(u = x^{2}\) and \(\frac{du}{dx}=2x\) into the formula, we get \(f^\prime(x)=-6\sin(x^{2})\cdot2x\).

Answer:

\(- 12x\sin(x^{2})\) (Option M)