CBSE Class 12 · 2013 · DifferentiationLong Answer

Differentiate the following function with respect to

x

(lo g x)^{x} + x^{l o g x}

Answer: $(lo g x)^{x} [\frac{1}{lo g x} + lo g (lo g x)] + \frac{2 lo g x}{x} x^{l o g x}$

Solution

Let

y = u + v

with

u = (lo g x)^{x}, v = x^{l o g x}

.
For

u

lo g u = x lo g (lo g x)

, differentiate:

\frac{d u}{d x} = (lo g x)^{x} [\frac{1}{lo g x} + lo g (lo g x)]

For

v

lo g v = (lo g x)^{2}

, differentiate:

\frac{d v}{d x} = x^{l o g x} \cdot \frac{2 lo g x}{x}

Add:

\frac{d y}{d x} = (lo g x)^{x} [\frac{1}{lo g x} + lo g (lo g x)] + x^{l o g x} \cdot \frac{2 lo g x}{x}

CBSE Class 12 Board · 2013 · Subjective