Question

if (f(x)=(cos(sqrt{x})-ln(x^{2}))^{3}), then (f(x)=)
a) ((-\frac{1}{2sqrt{x}}sin(sqrt{x})-\frac{2}{x})^{3})
b) (3(cos(sqrt{x})-ln(x^{2}))^{2}(-sin(sqrt{x})-\frac{1}{x}))
c) (3(cos(sqrt{x})-ln(x^{2}))^{2}(\frac{1}{2sqrt{x}}cos(sqrt{x})-\frac{2}{x}))
d) (3(cos(sqrt{x})-ln(x^{2}))^{2}(-\frac{1}{2sqrt{x}}sin(sqrt{x})-\frac{2}{x}))

Explanation:

Step1: Apply chain - rule

Let \(u = \sqrt{x}\), \(v=x^{2}\). The function \(y = 3\cos(\sqrt{x})-\ln(x^{2})\) can be differentiated term - by - term. The derivative of \(\cos(u)\) with respect to \(x\) is \(-\sin(u)\cdot u'\) and the derivative of \(\ln(v)\) with respect to \(x\) is \(\frac{v'}{v}\).
The derivative of \(\sqrt{x}=x^{\frac{1}{2}}\) is \(\frac{1}{2\sqrt{x}}\), and the derivative of \(x^{2}\) is \(2x\).
The derivative of \(3\cos(\sqrt{x})\) is \(3\times(-\sin(\sqrt{x}))\times\frac{1}{2\sqrt{x}}\), and the derivative of \(-\ln(x^{2})\) is \(-\frac{2x}{x^{2}}=-\frac{2}{x}\).

Step2: Combine the derivatives

\(y'=3\times(-\sin(\sqrt{x}))\times\frac{1}{2\sqrt{x}}-\frac{2}{x}=-\frac{3}{2\sqrt{x}}\sin(\sqrt{x})-\frac{2}{x}\)

Answer:

The derivative of \(y = 3\cos(\sqrt{x})-\ln(x^{2})\) is \(-\frac{3}{2\sqrt{x}}\sin(\sqrt{x})-\frac{2}{x}\), but since the options are not clearly readable in the provided image, we have shown the general derivative - finding process. If you can clarify the options, we can further match the answer.