Question

If$y=\cos (3{\cos }^{-1}x)$, then $\frac{d^{3}y}{d{x}^3}$is equal to

• A. $24$
• B. $27$
• C. $3–12x^2$
• D. $-24$

Answer 1

The correct option is A

$24$

Explanation for the correct option:

Step 1: Simplifying the given function:

Given that,

$y=\cos (3{\cos }^{-1}x)$

$\begin{array}{rcl}y& =& 4{(\cos {\cos }^{-1}x)}^3–3\cos {\cos }^{-1}x\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{3}\mathbf{x}\mathbf{}\mathbf{=}\mathbf{}\mathbf{4}\mathbf{}\mathbf{c}\mathbf{o}{\mathbf{s}}^{\mathbf{3}}\mathbf{x}\mathbf{}\mathbf{–}\mathbf{}\mathbf{3}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{x}\mathbf{]}\\ & =& 4x^3–3x\end{array}$

Step 2: Find the value of $\frac{d^{3}y}{d{x}^3}$:

Differentiate $y$ with respect to $x$,

$\frac{dy}{dx}=12x^2-3$

Differentiate $\frac{dy}{dx}$ with respect to $x$,

$\frac{d^{2}y}{d{x}^2}=24x$

Differentiate $\frac{d^{2}y}{d{x}^2}$ with respect to $x$:

$\frac{d^{3}y}{d{x}^3}=24$

Hence, the correct option is A.