Question

The derivative of ${\sin }^{3}x$ with respect to $\ln x$ is

[2 marks]

• A. $x{\sin }^{2}x\cos x$
• B. $3x{\sin }^{2}x\cos x$
• C. $3x{\sin }^{2}x$
• D. $3{\sin }^{2}x\cos x$

Answer 1

The correct option is B $3x{\sin }^{2}x\cos x$

Let $y={\sin }^{3}x$ and $z=\ln x$
$\frac{dy}{dx}=3{\sin }^{2}x\cos x$
and $\frac{dz}{dx}=\frac{1}{x}$
Now, $\frac{dy}{dz}=\frac{dy/dx}{dz/dx}=\frac{3{\sin }^{2}x\cos x}{1/x}$
$\therefore \frac{dy}{dz}=3x{\sin }^{2}x\cos x$