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  • Provide solution for RD Sharma maths class 12 chapter Differentiation exercise 10.2 question 45

Provide solution for RD Sharma maths class 12 chapter Differentiation exercise 10.2 question 45

Answers (1)

Answer: 2 x+\frac{2 e^{x}}{\sqrt{x^{4}-1}}

Hint: you must know the rules of solving derivative of polynomials

Given: \frac{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}{\sqrt{x^{2}+1}-\sqrt{x^{2}-1}}

Solution:

Let  y=\frac{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}{\sqrt{x^{2}+1}-\sqrt{x^{2}-1}}

By rationalizing,

\frac{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}{\sqrt{x^{2}+1}-\sqrt{x^{2}-1}} \times \frac{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}

\Rightarrow \frac{\left(\sqrt{x^{2}+1}+\sqrt{\left.x^{2}-1\right)}\right)^{2}}{\left(\left(\sqrt{\left.x^{2}+1\right)^{2}}-\left(\sqrt{\left.\left.x^{2}-1\right)^{2}\right)}\right.\right.\right.}

\Rightarrow \frac{\left(\sqrt{x^{2}+1}\right)^{2}+\left(\sqrt{x^{2}-1}\right)^{2}+2\left(\sqrt { x ^ { 2 } + 1 ) } \left(\sqrt{\left.x^{2}-1\right)}\right.\right.}{x^{2}+1-x^{2}+1}

=\frac{x^{2}+1+x^{2}-1+2 \sqrt{x^{4}-1}}{2}

=x^{2}+\sqrt{x^{4}-1}

Now, let  y=x^{2}+\sqrt{x^{4}-1}

Now, differentiate

\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(x^{2}+\sqrt{x^{4}-1}\right)

\begin{aligned} &\Rightarrow 2 x+\frac{1}{2 \sqrt{x^{4}-1}} \times\left(4 x^{3}\right) \\\\ &\Rightarrow 2 x+\frac{2 x^{3}}{\sqrt{x^{4}-1}} \end{aligned}

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