Name: Continuity Theorem
Uploaded: 2019-07-19
Duration: 5 min 49 s
Description: 5 is greater than 3, but 3 is not greater than 5. Similarly, if a function is differentiable, it is continuous, but the converse is not true. Watch this vide...

Question

Find the derivative of $y = \frac{x e^{x}}{x + e^{x}}$ w.r.t $x$ .

Accepted Answer

Given, $y = \frac{x e^{x}}{x + e^{x}}$

Differentiate $y$ w.r.t $x$ on each side, we get: $\frac{d y}{d x} = \frac{d}{d x} [\frac{x e^{x}}{x + e^{x}}]$

$\Rightarrow \frac{d y}{d x} = \frac{(x + e^{x}) \frac{d}{d x} (x e^{x}) - (x e^{x}) \frac{d}{d x} (x + e^{x})}{{(x + e^{x})}^{2}} [∵ \frac{d}{d x} (\frac{u}{v}) = \frac{v \cdot \frac{d u}{d x} - u \cdot \frac{d v}{d x}}{v^{2}}]$

$\Rightarrow \frac{d y}{d x} = \frac{(x + e^{x}) (x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x]) - (x e^{x}) (\frac{d}{d x} [x] + \frac{d}{d x} [e^{x}])}{{(x + e^{x})}^{2}} [∵ \frac{d}{d x} (u v) = u \cdot \frac{d v}{d x} + v \cdot \frac{d u}{d x}]$

$\Rightarrow \frac{d y}{d x} = \frac{(x + e^{x}) (x e^{x} + e^{x}) - (x e^{x}) (1 + e^{x})}{{(x + e^{x})}^{2}} [∵ \frac{d (e^{x})}{d x} = e^{x}]$

$\Rightarrow \frac{d y}{d x} = \frac{x^{2} e^{x} + x e^{2 x} + x e^{x} + e^{2 x} - x e^{2 x} - x e^{x}}{{(x + e^{x})}^{2}}$

$\Rightarrow \frac{d y}{d x} = \frac{x^{2} e^{x} + e^{2 x}}{{(x + e^{x})}^{2}}$

$\Rightarrow \frac{d y}{d x} = \frac{e^{x} [x^{2} + e^{x}]}{{(x + e^{x})}^{2}}$

Hence, the derivative of $y = \frac{x e^{x}}{x + e^{x}}$ is $\frac{e^{x} [x^{2} + e^{x}]}{{(x + e^{x})}^{2}}$ .

Find the derivative of $y = \frac{x e^{x}}{x + e^{x}}$ w.r.t $x$ .

Important Questions on Diffrentiation

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Find the derivative of y=xexx+ex w.r.t x.

Important Questions on Diffrentiation

Learn and Prepare for any exam you want !

Find the derivative of $y = \frac{x e^{x}}{x + e^{x}}$ w.r.t $x$ .