CBSE Class 12 · 2007 · DifferentiationLong Answer

Differentiate

sin (x^{2} + 1)

with respect to

x

from first principles.

Answer: $2 x cos (x^{2} + 1)$

Solution

Let

f (x) = sin (x^{2} + 1)

. Then:

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

Apply

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

. Here

\frac{C - D}{2} = \frac{2 x h + h ^{2}}{2}

and

\frac{C + D}{2} \to x^{2} + 1

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

Using

\frac{sin θ}{θ} \to 1

with

θ = \frac{h ( 2 x + h )}{2}

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

\frac{d}{d x} sin (x^{2} + 1) = 2 x cos (x^{2} + 1)

CBSE Class 12 Board · 2007 · Subjective