Name: Differentiation of Exponential Functions
Uploaded: 2019-07-19
Duration: 12 min 3 s
Description: To know how to find the derivative of exponential functions or multiple functions containing the exponential term. Let's watch this video and explore!

Question

Differentiate

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

with respect to

x

.

Accepted Answer

The given function is $\frac{\sqrt{x^{2} + 1} + \sqrt{x^{2} - 1}}{\sqrt{x^{2} + 1} - \sqrt{x^{2} - 1}}$ .

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

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

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

$y = \frac{x^{2} + 1 + x^{2} - 1 + 2 \sqrt{{(x^{2})}^{2} - 1^{2}}}{x^{2} + 1 - (x^{2} - 1)} [∵ {(a + b)}^{2} = a^{2} + b^{2} + 2 a b]$

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

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

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

On differentiating both sides with respect to $x$ , we get

$\frac{d}{d x} (y) = \frac{d}{d x} (x^{2} + \sqrt{x^{4} - 1})$

$\frac{d y}{d x} = \frac{d}{d x} (x^{2}) + \frac{d}{d x} (\sqrt{x^{4} - 1})$ [Using addition rule ]

$\frac{d y}{d x} = 2 x + \frac{1}{2 \sqrt{x^{4} - 1}} \cdot \frac{d}{d x} (x^{4} - 1)$ [Using chain rule]

$\frac{d y}{d x} = 2 x + \frac{1}{2 \sqrt{x^{4} - 1}} \cdot (4 x^{3} - 0) [∵ \frac{d}{d x} \sqrt{x} = \frac{1}{2 \sqrt{x}}; \frac{d}{d x} x^{n} = n x^{n - 1}]$

Thus, $\frac{d y}{d x} = 2 x + \frac{2 x^{3}}{\sqrt{x^{4} - 1}}$

Differentiate $\frac{\sqrt{x^{2} + 1} + \sqrt{x^{2} - 1}}{\sqrt{x^{2} + 1} - \sqrt{x^{2} - 1}}$ with respect to $x$ .

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Differentiate x2+1+x2-1x2+1-x2-1 with respect to x.

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Differentiate $\frac{\sqrt{x^{2} + 1} + \sqrt{x^{2} - 1}}{\sqrt{x^{2} + 1} - \sqrt{x^{2} - 1}}$ with respect to $x$ .