If y=tan^{-1}({5-2tan{x}}{2+5tan{x}}) then what is dy/dx equal to?

Correct Answer is : −12x-{1}{2{x}}−2x1

Correct Answer is : −12x-{1}{2{x}}−2x1

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UPSC NDA II 2018 Mathematics Solved Paper

Question : 70 of 120

Marks: +1, -0

y=tan^{-1}(\frac{5-2tan\sqrt{x}}{2+5tan\sqrt{x}})

\frac{dy}{dx}

equal to?

Solution:

Concept:

Use the identity

\tan^{-1} a - \tan^{-1} b = \tan^{-1}\!\left( \dfrac{a - b}{1 + ab} \right)

and simplify the given expression.

Explanation:

Given

y = \tan^{-1}\!\left( \dfrac{5 - 2\tan\sqrt{x}}{2 + 5\tan\sqrt{x}} \right)

Factor

2

from numerator and denominator:

y = \tan^{-1}\!\left( \dfrac{2\left(\frac{5}{2} - \tan\sqrt{x}\right)}{2\left(1 + \frac{5}{2}\tan\sqrt{x}\right)} \right) = \tan^{-1}\!\left( \dfrac{\frac{5}{2} - \tan\sqrt{x}}{1 + \frac{5}{2}\tan\sqrt{x}} \right)

Now apply the subtraction identity with

a = \frac{5}{2}

and

b = \tan\sqrt{x}

y = \tan^{-1}\frac{5}{2} - \tan^{-1}(\tan\sqrt{x}) = \tan^{-1}\frac{5}{2} - \sqrt{x}

Differentiate with respect to

x

\frac{dy}{dx} = 0 - \frac{1}{2\sqrt{x}} = -\frac{1}{2\sqrt{x}}

Answer:

The derivative is

-\dfrac{1}{2\sqrt{x}}

, which corresponds to option A.

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