If y= ^{-1}≤ft(1+ 5 x/ 5 x), then the value of d y/d x is

Correct Answer is : −52-5/2−25

Correct Answer is : −52-5/2−25

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MHTCET 19 Apr 2026 Shift 1 Paper

Question : 123 of 150

Marks: +1, -0

y=\cot ^{-1}\left(\frac{1+\sin 5 x}{\cos 5 x}\right)

\frac{d y}{d x}

Solution:

Concept:

Use the trigonometric identity

\frac{1+\sin \theta}{\cos \theta} = \tan\left(\frac{\pi}{4}+\frac{\theta}{2}\right)

and then differentiate the resulting linear expression.

Explanation:

Let

\theta = 5x

. The given expression becomes

\frac{1+\sin \theta}{\cos \theta}

Using the standard identity,

\frac{1+\sin \theta}{\cos \theta} = \tan\left(\frac{\pi}{4}+\frac{\theta}{2}\right)

Substituting

\theta = 5x

, we get

y = \cot^{-1}\left[\tan\left(\frac{\pi}{4}+\frac{5x}{2}\right)\right]

Since

\cot^{-1}(\tan z) = \frac{\pi}{2} - z

, we obtain

y = \frac{\pi}{2} - \left(\frac{\pi}{4}+\frac{5x}{2}\right) = \frac{\pi}{4} - \frac{5x}{2}

Differentiating with respect to

x

\frac{d y}{d x} = -\frac{5}{2}

Thus, the required derivative is constant and equals

-\frac{5}{2}

Answer:

Option D:

-\frac{5}{2}

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