Differentiate from first principle-v -sin 3x-1

Given: f(x)=√(sin (3x+1)) The derivative of a function f(x) is defined as: f'(x) = lim_h→ 0f(x+h) f(x)h Putting f(x) in above expression, we get: f'(x) = l ...

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Byju's Answer

Standard XII

Mathematics

Formation of a Differential Equation from a General Solution

Differentiate...

Question

Differentiate from first principle:

$(v) \sqrt{sin (3 x + 1)}$

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Solution

Given:

$f (x) = \sqrt{sin (3 x + 1)}$

The derivative of a function $f (x)$ is defined as:

$f^{'} (x) = lim h \to 0 \frac{f (x + h) - f (x)}{h}$

Putting $f (x)$ in above expression, we get:

$f^{'} (x) = lim h \to 0 \frac{\sqrt{sin (3 (x + h) + 1) - \sqrt{sin (3 x + 1)}}}{h}$

Rationalizing the numerator, we get:

$f^{'} (x) = lim h \to 0 \frac{\sqrt{sin (3 (x + h) + 1) - \sqrt{sin (3 x + 1)}}}{h} \times \frac{\sqrt{sin (3 x + 3 h + 1)} + \sqrt{sin (3 x + 1)}}{\sqrt{sin (3 x + 3 h + 1)} + \sqrt{sin (3 x + 1)}}$

$\Rightarrow f^{'} (x) = {lim}_{h \to 0} \frac{sin (3 x + 3 h + 1) - sin (3 x + 1)}{h (\sqrt{sin (3 x + 3 h + 1)} + \sqrt{sin (3 x + 1)})}$

Applying the formula,

$sin c - sin d = 2 cos (\frac{c + d}{2}) sin (\frac{c - d}{2}),$ we get

$\Rightarrow f^{'} (x) = {lim}_{h \to 0} \frac{2 cos (\frac{3 x + 3 h + 1 + 3 x + 1}{2}) sin (\frac{3 x + 3 h + 1 - 3 x - 1}{2})}{h \sqrt{(sin (3 x + 3 h + 1) + \sqrt{sin (3 x + 1)})}}$

$\Rightarrow f^{'} (x) = {lim}_{h \to 0} \frac{2 cos (\frac{6 x + 3 h + 2}{2}) sin \frac{3 h}{2}}{h (\sqrt{sin (3 x + 3 h + 1)} + \sqrt{sin (3 x + 1)})}$

$\Rightarrow f^{'} (x) = {lim}_{h \to 0} \frac{2 cos (\frac{6 x + 3 h + 2}{2})}{(\sqrt{sin (3 x + 3 h + 1)} + \sqrt{sin (3 x + 1)})} . \frac{sin \frac{3 h}{2}}{\frac{3 h}{2}} . \frac{3}{2}$

$\Rightarrow f^{'} (x) = 3 \frac{cos (\frac{6 x + 2}{2})}{(\sqrt{sin (3 x + 1)} + \sqrt{sin (3 x + 1)})}$

$[∵ lim h \to 0 \frac{sin h}{h} = 1]$

$\Rightarrow f^{'} (x) = \frac{3 cos (3 x + 1)}{2 \sqrt{sin (3 x + 1)}}$

Suggest Corrections

Differentiate from first principle: (v) √sin(3x+1)

Differentiate from first principle:

$(v) \sqrt{sin (3 x + 1)}$