Question

Differentiate $ {x^{{{\sin }^{ - 1}}x}} $ w.r.t $ {\sin ^{ - 1}}x $ .
A. $ {x^{{{\sin }^{ - 1}}x}}\left[ {\log x + {{\sin }^{ - 1}}x \cdot \dfrac{{\sqrt {(1 - {x^2})} }}{x}} \right] $
B. $ - {x^{{{\sin }^{ - 1}}x}}\left[ {\log x + {{\sin }^{ - 1}}x \cdot \dfrac{{\sqrt {(1 - {x^2})} }}{x}} \right] $
C. $ {x^{{{\sin }^{ - 1}}x}}\left[ {\log x + {{\sin }^{ - 1}}x \cdot \dfrac{{\sqrt {(1 + {x^2})} }}{x}} \right] $
D. $ - {x^{{{\sin }^{ - 1}}x}}\left[ {\log x + {{\sin }^{ - 1}}x \cdot \dfrac{{\sqrt {(1 + {x^2})} }}{x}} \right] $

Answer 1

Hint: As we can see that we have to differentiate the given function with respect to $ {\sin ^{ - 1}}x $ . In this question we will assume the expressions into smaller variables to solve the easier problem. Then we take the $ \log $ to the both sides of the equation. We should know that the formula of log is that if there is $ \log {(a)^b} $ , then it can be written as $ b\log a $ .

Complete step by step solution:
Let us assume that $ y = {x^{{{\sin }^{ - 1}}x}} $ and the power i.e. $ z = {\sin ^{ - 1}}x \Rightarrow z = \dfrac{1}{{\sin }}x $ . It can be written as $ \sin z = x $ .
Now we have to find $ \dfrac{{dy}}{{dz}} $ . We have $ y = {(\sin z)^z} $ , by taking log on the both sides we have $ \log y = \log {(\sin z)^z} $ .
By applying the above logarithm formula we have: $ \log y = z\log \sin z $ .
Now we will differentiate it w.r.t $ z $ $ (z = {\sin ^{ - 1}}x) $ .
We can write it as $ \dfrac{1}{y}\dfrac{{dy}}{{dz}} = \log \sin z + z\dfrac{{\cos z}}{{\sin z}} $ , by transferring $ y $ to the other side of the equation: $ \dfrac{{dy}}{{dz}} = y\left( {\log \sin z + z\dfrac{{\cos z}}{{\sin z}}} \right) $ .
If we have $ \sin z = x $ , then we can write the value of $ \cos z = \sqrt {1 - {{\sin }^2}z} = \sqrt {1 - {x^2}} $ .
By putting the values back in the equation we have: $ {x^{{{\sin }^{ - 1}}x}}\left[ {\log x + {{\sin }^{ - 1}}x \cdot \dfrac{{\sqrt {(1 - {x^2})} }}{x}} \right] $ .
Hence the correct option is (a) $ {x^{{{\sin }^{ - 1}}x}}\left[ {\log x + {{\sin }^{ - 1}}x \cdot \dfrac{{\sqrt {(1 - {x^2})} }}{x}} \right] $ .
So, the correct answer is “Option A”.

Note: We should note that to find the value of $ \cos z $ , we have used an identity which is $ {\sin ^2}\theta + {\cos ^2}\theta = 1 $ . So to find the value of $ \cos \theta = \sqrt {1 - {{\sin }^2}\theta } $ . We should solve carefully to avoid any calculation mistakes and we should be fully aware of the trigonometric identities and how to differentiate them.