Differentiation Question 430
Question: The first derivative of the function $ [ {{\cos }^{-1}}( \sin \sqrt{\frac{1+x}{2}} )+x^{x} ] $ with respect to x at x = 1 is
[MP PET 1998]
Options:
A) $ \frac{3}{4} $
B) 0
C) $ \frac{1}{2} $
D) $ -\frac{1}{2} $
Show Answer
Answer:
Correct Answer: A
Solution:
$ f(x)={{\cos }^{-1}}[ \cos ( \frac{\pi }{2}-\sqrt{\frac{1+x}{2}} ) ]+x^{x} $
$ f(x)=\frac{\pi }{2}-\sqrt{\frac{1+x}{2}}+x^{x} $
$ \therefore f’(x)=-\frac{1}{\sqrt{2}}.\frac{1}{2\sqrt{1+x}}+x^{x}(1+\log x) $
$ f’(1)=-\frac{1}{4}+1=\frac{3}{4} $ .
BETA
