Differentiate $\sin\sqrt{2x}$ with respect to $x$ from first principles.

$\dfrac{\cos\sqrt{2x}}{\sqrt{2x}}$

CBSE Class 12 · 2007 · DifferentiationLong Answer

Differentiate

sin 2 x

with respect to

x

from first principles.

Answer: $\frac{cos 2 x}{2 x}$

Let

f (x) = sin 2 x

. Then:

f^{'} (x) = h \to 0 lim \frac{sin 2 ( x + h ) - sin 2 x}{h}

Use

sin C - sin D = 2 cos \frac{C + D}{2} sin \frac{C - D}{2}

and rationalise

2 (x + h) - 2 x = \frac{2 h}{2 ( x + h ) + 2 x}

f^{'} (x) = cos 2 x \cdot h \to 0 lim \frac{2 ( x + h ) - 2 x}{h} = cos 2 x \cdot \frac{2}{2 2 x}

\frac{d}{d x} sin 2 x = \frac{cos 2 x}{2 x}

CBSE Class 12 Board · 2007 · Subjective