Derivative of Cosine Square Root of X

In trigonometric Differentiation most of the examples based on sine square roots function, we will discuss in detail derivative of cosine square root of x function and its related examples. It can be proved by definition of differentiation.

Consider the function of the form

y = f\left( x \right) = \cos \sqrt x

We can prove this with the help of definition of differentiation, we have

\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {x + \Delta x} \right) - f\left( x \right)}}{{\Delta x}}\,\,\,\,{\text{ - - - }}\left( {\text{i}} \right)

Putting the value of function in equation (i), we get

\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\cos \sqrt {x + \Delta x} - \cos \sqrt x }}{{\Delta x}}


Using formula from trigonometry

\cos A - \cos B = - 2\sin \left( {\frac{{A + B}}{2}} \right)\sin \left( {\frac{{A - B}}{2}} \right)


\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{ - 2\sin \left( {\frac{{\sqrt {x + \Delta x} + \sqrt x }}{2}} \right)\sin \left( {\frac{{\sqrt {x + \Delta x} - \sqrt x }}{2}} \right)}}{{\Delta x}}

Now consider the relation

\left( {\sqrt {x + \Delta x} + \sqrt x } \right)\left( {\sqrt {x + \Delta x} - \sqrt x } \right) = {\left( {\sqrt {x + \Delta x} } \right)^2} - {\left( {\sqrt x } \right)^2} = \Delta x


\begin{gathered} \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{ - 2\sin \left( {\frac{{\sqrt {x + \Delta x} + \sqrt x }}{2}} \right)\sin \left( {\frac{{\sqrt {x + \Delta x} - \sqrt x }}{2}} \right)}}{{\left( {\sqrt {x + \Delta x} + \sqrt x } \right)\left( {\sqrt {x + \Delta x} - \sqrt x } \right)}} \\ \Rightarrow \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{ - 2\sin \left( {\frac{{\sqrt {x + \Delta x} + \sqrt x }}{2}} \right)}}{{\left( {\sqrt {x + \Delta x} + \sqrt x } \right)}} \times \frac{{\sin \left( {\frac{{\sqrt {x + \Delta x} - \sqrt x }}{2}} \right)}}{{\left( {\sqrt {x + \Delta x} - \sqrt x } \right)}} \\ \Rightarrow \frac{{dy}}{{dx}} = - \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\sin \left( {\frac{{\sqrt {x + \Delta x} + \sqrt x }}{2}} \right)}}{{\left( {\sqrt {x + \Delta x} + \sqrt x } \right)}} \times \frac{{\sin \left( {\frac{{\sqrt {x + \Delta x} - \sqrt x }}{2}} \right)}}{{\left( {\frac{{\sqrt {x + \Delta x} - \sqrt x }}{2}} \right)}} \\ \Rightarrow \frac{{dy}}{{dx}} = - \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\sin \left( {\frac{{\sqrt {x + \Delta x} + \sqrt x }}{2}} \right)}}{{\left( {\sqrt {x + \Delta x} + \sqrt x } \right)}}\mathop {\lim }\limits_{\Delta x \to 0} \frac{{\sin \left( {\frac{{\sqrt {x + \Delta x} - \sqrt x }}{2}} \right)}}{{\left( {\frac{{\sqrt {x + \Delta x} - \sqrt x }}{2}} \right)}} \\ \end{gathered}

Consider \frac{{\sqrt {x + \Delta x} - \sqrt x }}{2} = u, as \Delta x \to 0, then u \to 0

\begin{gathered} \frac{{dy}}{{dx}} = - \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\sin \left( {\frac{{\sqrt {x + \Delta x} + \sqrt x }}{2}} \right)}}{{\left( {\sqrt {x + \Delta x} + \sqrt x } \right)}}\mathop {\lim }\limits_{u \to 0} \frac{{\sin \left( u \right)}}{{\left( u \right)}} \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{{\sin \left( {\frac{{\sqrt {x + 0} + \sqrt x }}{2}} \right)}}{{\left( {\sqrt {x + 0} + \sqrt x } \right)}}\left( 1 \right) \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{{\sin \sqrt x }}{{2\sqrt x }} \\ \end{gathered}

Example: Find the derivative of

y = f\left( x \right) = \cos \sqrt {{x^2} + 1}

We have the given function as

y = \cos \sqrt {{x^2} + 1}

Differentiation with respect to variable x, we get

\frac{{dy}}{{dx}} = \frac{d}{{dx}}\cos \sqrt {{x^2} + 1}

Using the rule, \frac{d}{{dx}}\cos \sqrt x = - \frac{{\sin \sqrt x }}{{2\sqrt x }}, we get

\begin{gathered} \frac{{dy}}{{dx}} = - \frac{{\sin \sqrt {{x^2} + 1} }}{{2\sqrt {{x^2} + 1} }}\frac{d}{{dx}}\left( {{x^2} + 1} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{{\sin \sqrt {{x^2} + 1} }}{{2\sqrt {{x^2} + 1} }}\left( {2x} \right) \\ \Rightarrow \frac{{dy}}{{dx}} = - \frac{{x\sin \sqrt {{x^2} + 1} }}{{\sqrt {{x^2} + 1} }} \\ \end{gathered}