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 following function:

(\sqrt{x} + \frac{1}{x}) (x - \frac{1}{\sqrt{x}})

Accepted Answer

Given: $(\sqrt{x} + \frac{1}{x}) (x - \frac{1}{\sqrt{x}})$

We know that, from chain rule $\frac{d}{d x} (u v) = u \frac{d}{d x} v + v \frac{d}{d x} u$

Here, $u = (x^{\frac{1}{2}} + \frac{1}{x}), v = (x - x^{\frac{- 1}{2}})$

Now, $\frac{d}{d x} (x^{\frac{1}{2}} + x^{- 1}) (x - x^{\frac{- 1}{2}})$

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

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

$= x^{\frac{1}{2}} + x^{- 1} + \frac{1}{2} x^{- 1} + \frac{1}{2} x^{\frac{- 5}{2}} + \frac{1}{2} x^{\frac{1}{2}} - x^{- 1} - \frac{1}{2} x^{- 1} + x^{\frac{- 5}{2}}$

$= \frac{3}{2} x^{\frac{1}{2}} + \frac{3}{2} x^{\frac{- 5}{2}}$

Therefore, $\frac{d}{d x} ((\sqrt{x} + \frac{1}{x}) (x - \frac{1}{\sqrt{x}})) = \frac{3}{2} (x^{\frac{1}{2}} + x^{\frac{- 5}{2}})$

Differentiate the following function: $(\sqrt{x} + \frac{1}{x}) (x - \frac{1}{\sqrt{x}})$

Important Questions on Differentiation

Learn and Prepare for any exam you want !

Differentiate the following function:x+1xx-1x

Important Questions on Differentiation

Learn and Prepare for any exam you want !

Differentiate the following function: $(\sqrt{x} + \frac{1}{x}) (x - \frac{1}{\sqrt{x}})$