Question

If \[y = \cos \left( {3{{\cos }^{ - 1}}x} \right)\] then \[\dfrac{{{d^3}y}}{{d{x^3}}}\] is
A. 24
B. 27
C. \[3 - 12{x^2}\]
D. -24

Answer 1

Hint: Given function is combination of trigonometric and algebraic terms. we have to find its third order derivative. But we cannot simply jump on it, we will find first derivative, then second derivative and then third derivative. Generally up to the third order derivative we will get a constant number. But let’s check in this case.

Complete step by step answer:
Given function is,
\[y = \cos \left( {3{{\cos }^{ - 1}}x} \right)\]
We know that, \[\cos 3\theta = 4{\cos ^3}\theta - 3\cos \theta \]
So the above equation can be written as,
\[y = 4{\left( {\cos .{{\cos }^{ - 1}}x} \right)^3} - 3\cos {\cos ^{ - 1}}x\]
We know that \[\cos .{\cos ^{ - 1}}x = x\]
\[y = 4{x^3} - 3x\]
Now taking the derivatives,
\[\dfrac{{dy}}{{dx}} = 4 \times 3{x^2} - 3\]
On multiplying we get,
\[\dfrac{{dy}}{{dx}} = 12{x^2} - 3\]
Taking the second derivative,
\[\dfrac{{{d^2}y}}{{d{x^2}}} = 12 \times 2x = 24x\]
Now the last third derivative,
\[\therefore \dfrac{{{d^3}y}}{{d{x^3}}} = 24\]

Thus the correct option is A.

Note:Note that, the given function is given in trigonometric form but we have to convert it in the simplest form. For that we used the triple angle formula. We have found the third order derivative. If asked it can be extended upto some bigger order also. But the step at which we get a constant is the last one or we can say that the derivative of constant is zero.