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provide solution for RD Sharma maths class 12 chapter Differentiation exercise 10.2 question 33

Answers (1)

Answer:\frac{e^{x}}{1+e^{2 x}}

Hint: You must know about the rules of solving derivative of Inverse trigonometric function and exponential

Given: \tan ^{-1}\left(e^{x}\right)

Solution:

Let  y=\tan ^{-1}\left(e^{x}\right)

Differentiate with respect to x,

\begin{aligned} &\frac{d y}{d x}=\frac{d}{d x}\left[\tan ^{-1}\left(e^{x}\right)\right] \\ \\&\frac{d y}{d x}=\frac{1}{1+\left(e^{x}\right)^{2}} \frac{d}{d x}\left(e^{x}\right) \end{aligned}

\begin{aligned} &\frac{d y}{d x}=\frac{1}{1+e^{2 x}} \times e^{x} \\\\ &\frac{d y}{d x}=\frac{e^{x}}{1+e^{2 x}} \end{aligned}

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