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

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

CBSE Class 12 · 2005 · DifferentiationLong Answer

Differentiate

sin x

with respect to

x

from first principles.

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

Let

f (x) = sin x

. By definition:

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

Apply

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

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

Rationalise

x + h - x = \frac{h}{x + h + x}

, and use

\frac{sin θ}{θ} \to 1

f^{'} (x) = cos x \cdot h \to 0 lim \frac{x + h - x}{h} = cos x \cdot \frac{1}{2 x}

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

CBSE Class 12 Board · 2005 · Subjective