Question

Differential coefficient of \[\sqrt {\sec \sqrt x } \] is

Answer 1

Hint: Differential coefficient of the given function is nothing but finding the derivative of the function with respect to x. We will find the derivative of the function given above step by step. Like first the derivative of the root of a function and then the derivative of the function inside. Also we will try to simplify the equation as much as possible.

Complete step-by-step answer:
Given that, \[\sqrt {\sec \sqrt x } \]
Now we will differentiate the function above,
\[\dfrac{d}{{dx}}\sqrt {\sec \sqrt x } \]
Now we know that, \[\dfrac{d}{{dx}}\sqrt x = \dfrac{1}{{2\sqrt x }}\]
So the function above can be written as,
\[\dfrac{d}{{dx}}\sqrt {\sec \sqrt x } = \dfrac{1}{{2\sqrt {\sec \sqrt x } }}.\dfrac{d}{{dx}}\left[ {\sec \sqrt x } \right]\]
Now we know that derivative of \[\sec x\] is \[\sec x.\tan x\]
\[\dfrac{d}{{dx}}\sqrt {\sec \sqrt x } = \dfrac{1}{{2\sqrt {\sec \sqrt x } }}.\sec \sqrt x \tan \sqrt x .\dfrac{1}{{2\sqrt x }}\]
This is the complete derivative.
Now we will simplify the answer as much as possible.
\[\dfrac{d}{{dx}}\sqrt {\sec \sqrt x } = \dfrac{{\sec \sqrt x \tan \sqrt x .\dfrac{1}{{2\sqrt x }}}}{{2\sqrt {\sec \sqrt x } }}\]
The denominator of the last fraction will be shifted to the bottom,
\[\dfrac{d}{{dx}}\sqrt {\sec \sqrt x } = \dfrac{{\sec \sqrt x \tan \sqrt x }}{{2 \times 2\sqrt x .\sqrt {\sec \sqrt x } }}\]
Now the denominator will be simplified,
\[\dfrac{d}{{dx}}\sqrt {\sec \sqrt x } = \dfrac{{\sec \sqrt x \tan \sqrt x }}{{4\sqrt x .\sqrt {\sec \sqrt x } }}\]
Now tan function can be written in the form of sin and cos function.
\[\dfrac{d}{{dx}}\sqrt {\sec \sqrt x } = \dfrac{1}{{4\sqrt x }}.\dfrac{{\sec \sqrt x .\dfrac{{\sin \sqrt x }}{{\cos \sqrt x }}}}{{\sqrt {\sec \sqrt x } }}\]
Now the cos function is the reciprocal of the sec function.
\[\dfrac{d}{{dx}}\sqrt {\sec \sqrt x } = \dfrac{1}{{4\sqrt x }}.\dfrac{{\sec \sqrt x .\sec \sqrt x \sin \sqrt x }}{{\sqrt {\sec \sqrt x } }}\]
We can cancel one sec function with the root in the denominator.
\[\dfrac{d}{{dx}}\sqrt {\sec \sqrt x } = \dfrac{1}{{4\sqrt x }}.\sec \sqrt x .\sqrt {\sec \sqrt x } .\sin \sqrt x \]
Now we can write the sec function as,
\[\dfrac{d}{{dx}}\sqrt {\sec \sqrt x } = \dfrac{1}{{4\sqrt x }}{\left( {\sec \sqrt x } \right)^{\dfrac{3}{2}}}\sin \sqrt x \]
This is a simplified answer.

Note: Here note that finding the derivative is not that difficult. But writing the simplified form is a tricky task. Maybe the options might differ from the answer we have obtained but that won’t mean the answer is wrong. It might be like the functions are arranged in some other way. So be careful how you proceed with your answer.
Also note that, always when there are functions in between then do find the derivative step by step so that not a single function is missed. Like here first the derivative of root and then that of the function inside. Also after finding the derivative of the function insides we have one more root there, that might get escaped. Remember the derivative of a constant is only zero.