Question

Find the differential coefficient of $\sqrt {\tan \sqrt x }$ .

Answer 1

Differential coefficient of a function is another term for the derivative of that function. So, we will differentiate the given function step by step and then we will proceed towards the final solution.

Complete step-by-step answer:
Let the given function be denoted by $y = \sqrt {\tan \sqrt x }$
Now, we will differentiate the given equation with respect to x. The differentiation will be done step by step. On differentiating the equation with respect to x, we get,
$\dfrac{{dy}}{{dx}} = \dfrac{{d(\sqrt {\tan \sqrt x } )}}{{dx}}$
=$\dfrac{1}{{2\sqrt {\tan \sqrt x } }}{\sec ^2}\sqrt x \dfrac{1}{{2\sqrt x }}$
In the above differentiation, we have used the differentiation of the functions as $\left( {\dfrac{{d\left( {\sqrt y } \right)}}{{dx}} = \dfrac{1}{{2\sqrt y }}} \right)$, and $\left( {\dfrac{{d\left( {\tan x} \right)}}{{dx}} = {{\sec }^2}x} \right)$.
Here, we have first differentiated the function$\sqrt {\tan \sqrt x }$after that we have differentiated the function $\tan \sqrt x$and lastly differentiated $\sqrt x$.
Simplifying the above equation, we get
$\dfrac{{dy}}{{dx}} = \dfrac{{{{\sec }^2}\sqrt x }}{{4\sqrt {x\tan \sqrt x } }}$
Therefore, the differential coefficient of $\sqrt {\tan \sqrt x }$will be $\dfrac{{{{\sec }^2}x}}{{4\sqrt {x\tan \sqrt x } }}$.

Additional Information: Differential basically refers to the infinitesimal differences or generally to the functions’ derivative.
A derivative is the change in a function while a differential coefficient is the change between two variables. Hence, we can say that the derivative is always a ratio of the differentials.
A coefficient is usually a constant quantity but the differential coefficient of any function is a constant function only and only if the given function is a linear function.

Note: In such questions where we are required to find the differential coefficient, we generally differentiate the given function as many times as the order of differential coefficient. For e.g., if nth differential coefficient is required, then we just differentiate the function n times.