CBSE Class 12 Math 2013 Solved Paper

Question : 14 of 29

Marks: +1, -0

Differentiate the following function with respect to x:

(logx)^π + x^{logπ}

Solution:

Let y =

(logx)^π + x^{logπ}

... (1)
Now let

y_1

(logx)^x

and yz =

x^{logx}

⇒ y =

y_1+y_2

... (2)
Differentiating (2) w.r.t. x,

{dy}/{dx}

{dy_1}/{dx}+{dy_2}/{dx}

... (3)
Now consider

y_1

(logx)^π

Taking log on both sides,
log

y_1

= x log (log x)
Differentiating w.r.t. x, we get

1/y_1 {dy_1}/{dx}

= x ×

1/{logx}

1/x

+ 1 - log (log x)
⇒

{dy_1}/{dx}

y_1(1/{logx} + log(logx))

⇒

{dy_1}/{dx}

log x^π (1/{logx} + log (logx))

... (4)
⇒ Now, consider

y_2

= (log x) (log x) =

(logx)^2

Differentiating w.r.t. x, we get

1/y_2 {dy_2}/{dx}

= 2 log x ×

1/x

⇒

{dy_2}/{dx}

y_2 ({2logx}/x)

x^{logx} ({2logx}/x)

... (5)
Using equations (3), (4) and (5), we get:

{dy}/{dx}

logx^π(1/{logx}+log(logx))

x^{logπ}({2logx}/x)

Go to Question:

Prev Question

Next Question