Question

Solve the given derivative, the expression is:
\[\dfrac{d}{{dx}}(\sin \sqrt x )\]?

Answer 1

Hint: The given question is of derivative in which the “sin” function is given, here we know the direct derivative of the sin function and will solve accordingly. One more property of derivative will be used here that is, in case of theta we know the direct derivative of “sin”, but here root of “x”, is given so we need to do the derivative of that also.

Formulae Used: Derivative of “sin” function is:
\[ \Rightarrow \dfrac{d}{{dx}}(\sin \theta ) = \cos \theta \]
Derivative of root of “x”:
\[ \Rightarrow \dfrac{d}{{dx}}(\sqrt x ) = \dfrac{1}{2}{x^{\dfrac{1}{2} - 1}} = \dfrac{1}{2}{x^{\dfrac{3}{2}}}\]

Complete step-by-step answer:
Here in the given question we need to find the derivative of the “sin” function, here we know that in place of the angle, we are given with root of “x”, to solve thios derivative we use the direct formulae, on solving we get:
\[ \Rightarrow \dfrac{d}{{dx}}(\sin \sqrt x ) = (\cos \sqrt x )\left( {\dfrac{1}{2}{x^{\dfrac{1}{2} - 1}}} \right) = \dfrac{1}{2}{x^{\dfrac{3}{2}}}\left( {\cos \sqrt x } \right)\]
Here we get the final solution of the derivative given.

Note:In order to solve the question of differentiation we need to remember the formulae of the derivative, without formulae one cannot solve the derivative questions. Here we used the rule for differentiating “sin” identity when any complex angle is given instead of theta.