Name: Rate of Change of Function: Example
Uploaded: 2019-07-19
Duration: 3 min 57 s
Description: In this video, we will use the rate of change to find the speed of the lower end of the ladder when it is sliding on the ground. Watch this video to know more.

Question

Find derivatives of the following functions from definition.

$cosec 3 x$

Accepted Answer

Let, $cosec 3 x$

We know that, from first principle of differentiation $f' (x) = \lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

Now, $f' (x) = \lim_{h \to 0} \frac{cosec (3 x + 3 h) - cosec 3 x}{h}$

$= \lim_{h \to 0} \frac{\{\frac{1}{\sin (3 x + 3 h)} - \frac{1}{\sin 3 x}\}}{h}$

$= \lim_{h \to 0} \frac{\sin 3 x - \sin (3 x + 3 h)}{h \cdot \sin 3 x \cdot \sin (3 x + 3 h)}$

$= \lim_{h \to 0} \frac{2 \cos (\frac{6 x + 3 h}{2}) \cdot \sin (\frac{- 3 h}{2})}{h . \sin 3 x \cdot \sin (3 x + 3 h)}$

$[∵ \sin α - \sin β = 2 \cos (\frac{α + β}{2}) \sin (\frac{α - β}{2})]$

$= \lim_{h \to 0} \frac{2 \cos (3 x + h) \sin (\frac{- 3 h}{2})}{\sin (3 x + 3 h) \times \frac{- 3 h}{2}} \times \frac{- 3}{2} \times \frac{1}{\sin 3 x}$

$= \lim_{h \to 0} \frac{2 \cos (3 x + h)}{\sin (3 x + 3 h)} \times \lim_{h \to 0} \frac{\sin (\frac{- 3 h}{2})}{\frac{- 3 h}{2}} \times \lim_{x \to 0} \frac{- 3}{2} \times \frac{1}{\sin 3 x}$

$= \frac{2 \cos (3 x)}{\sin 3 x} \times 1 \times - \frac{3}{2} \times \frac{1}{\sin 3 x} [∵ \lim_{x \to 0} \frac{\sin (x)}{x} = 1]$

$= 2 \cot 3 x \times - \frac{3}{2} \times cosec x = - 3 cosec 3 x \cdot \cot 3 x$

$∴ f^{'} (x) = - 3 cosec (3 x) \cot (3 x)$ .

Find derivatives of the following functions from definition.

$cosec 3 x$

Important Questions on Limit and Differentiation

Learn and Prepare for any exam you want !

Find derivatives of the following functions from definition. cosec3x

Important Questions on Limit and Differentiation

Learn and Prepare for any exam you want !

Find derivatives of the following functions from definition.

$cosec 3 x$