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 the function

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

w.r.t.

x

.

Accepted Answer

The given function is,

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

Let $u$ and $v$ be two functions of $x$ , such that

$u = x^{x \cos x}$ and $v = \frac{x^{2} + 1}{x^{2} - 1}$

Here, we get

$y = u + v$

$∴ \frac{d y}{d x} = \frac{d u}{d x} + \frac{d v}{d x}$

Let $u = x^{x \cos x}$

Taking $\log$ on both sides, we get

$\log u = \log x^{x \cos x}$

$\log u = x \cos x \log x$ $[∵ \log a^{b} = b \log a]$

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

$\frac{d}{d x} (\log u) = \frac{d}{d x} (x \cos x \log x)$

$\frac{1}{u} \frac{d u}{d x} = x \cos x \cdot \frac{d}{d x} (\log x) + \log x \frac{d}{d x} (x \cos x)$ $[$ Using product rule $]$

$\frac{1}{u} \frac{d u}{d x} = x \cos x \cdot \frac{1}{x} + \log x [x \cdot \frac{d}{d x} (\cos x) + \cos x \frac{d}{d x} (x)]$ $[∵ \frac{d}{d x} \log x = \frac{1}{x}; \frac{d}{d x} x^{n} = n x^{n - 1}]$

$\frac{1}{u} \frac{d u}{d x} = \cos x + \log x [x \cdot (- \sin x) + \cos x \cdot 1]$

$\frac{1}{u} \frac{d u}{d x} = \cos x + \log x (\cos x - x \sin x)$

$\frac{1}{u} \frac{d u}{d x} = \cos x + \log x \cos x - x \sin x \log x$

$\frac{1}{u} \frac{d u}{d x} = (1 + \log x) \cos x - x \sin x \log x$

$\frac{d u}{d x} = u \cdot ((1 + \log x) \cos x - x \sin x \log x)$ $[$ Cross multiply $u]$

$\frac{d u}{d x} = x^{x \cos x} \cdot ((1 + \log x) \cos x - x \sin x \log x)$ $[$ Substituting value from $u]$

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

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

$\frac{d}{d x} (v) = \frac{d}{d x} (\frac{x^{2} + 1}{x^{2} - 1})$

$\frac{d v}{d x} = \frac{(x^{2} - 1) \frac{d}{d x} (x^{2} + 1) - (x^{2} + 1) \frac{d}{d x} (x^{2} - 1)}{{(x^{2} - 1)}^{2}}$ $[$ Using quotient rule $]$

$\frac{d v}{d x} = \frac{(x^{2} - 1) [\frac{d}{d x} (x^{2}) + \frac{d}{d x} (1)] - (x^{2} + 1) [\frac{d}{d x} (x^{2}) - \frac{d}{d x} (1)]}{{(x^{2} - 1)}^{2}}$

$\frac{d v}{d x} = \frac{(x^{2} - 1) [2 x + 0] - (x^{2} + 1) [2 x - 0]}{{(x^{2} - 1)}^{2}}$ $[∵ \frac{d}{d x} x^{n} = n x^{n - 1}; \frac{d}{d x} (c) = 0]$

$\frac{d v}{d x} = \frac{(2 x^{3} - 2 x) - (2 x^{3} + 2 x)}{{(x^{2} - 1)}^{2}}$

$\frac{d v}{d x} = \frac{2 x^{3} - 2 x - 2 x^{3} - 2 x}{{(x^{2} - 1)}^{2}}$

$\frac{d v}{d x} = \frac{- 4 x}{{(x^{2} - 1)}^{2}}$

Then,

$\frac{d y}{d x} = \frac{d u}{d x} + \frac{d v}{d x}$

$\frac{d y}{d x} = x^{x \cos x} \cdot ((1 + \log x) \cos x - x \sin x \log x) - \frac{4 x}{{(x^{2} - 1)}^{2}}$ .

Differentiate the function $x^{x \cos x} + \frac{x^{2} + 1}{x^{2} - 1}$ w.r.t. $x$ .

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Differentiate the function xx cos x+x2+1x2-1 w.r.t. x.

Important Questions on Continuity and Differentiability

Learn and Prepare for any exam you want !

Differentiate the function $x^{x \cos x} + \frac{x^{2} + 1}{x^{2} - 1}$ w.r.t. $x$ .