Differentiate the function $y=(\sin x)^{x}+\sin^{-1}\sqrt{x}$ with respect to $x$.

$(\sin x)^{x}\big[x\cot x+\log\sin x\big]+\dfrac{1}{2\sqrt{x(1-x)}}$

CBSE Class 12 · 2009 · DifferentiationLong Answer

Differentiate the function

y = (sin x)^{x} + sin^{- 1} x

with respect to

x

Answer: $(sin x)^{x} [x cot x + lo g sin x] + \frac{1}{2 x ( 1 - x )}$

Write

y = u + v

with

u = (sin x)^{x}, v = sin^{- 1} x

.
For

u

lo g u = x lo g sin x

, differentiate:

\frac{d u}{d x} = (sin x)^{x} [x cot x + lo g sin x]

For

v

\frac{d v}{d x} = \frac{1}{1 - x} \cdot \frac{1}{2 x} = \frac{1}{2 x ( 1 - x )}

.
Add:

\frac{d y}{d x} = (sin x)^{x} [x cot x + lo g sin x] + \frac{1}{2 x ( 1 - x )}

CBSE Class 12 Board · 2009 · Subjective