Examples of Trigonometric Differentiation

Example:
Differentiate \frac{{\sqrt {\sin {\text{x}}} }}{{\sin \sqrt {\text{x}} }}with respect to ‘x’.

Solution:
Let {\text{y}} = \frac{{\sqrt {\sin {\text{x}}} }}{{\sin \sqrt {\text{x}} }}

Differentiate w.r.t ‘x’

\begin{gathered} \frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\text{d}}}{{{\text{dx}}}}\left[ {\frac{{\sqrt {\sin {\text{x}}} }}{{\sin \sqrt {\text{x}} }}} \right] \\ \Rightarrow \frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\sin \sqrt {\text{x}} \frac{{\text{d}}}{{{\text{dx}}}}\sqrt {\sin {\text{x}}} - \sqrt {\sin {\text{x}}} \frac{{\text{d}}}{{{\text{dx}}}}\sin \sqrt {\text{x}} }}{{{{\left( {\sin \sqrt {\text{x}} } \right)}^2}}} \\ \Rightarrow \frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\sin \sqrt {\text{x}} \left[ {\frac{1}{{2\sqrt {\sin {\text{x}}} }} \cdot \frac{{\text{d}}}{{{\text{dx}}}}\sin {\text{x}}} \right] - \sqrt {\sin {\text{x}}} \cdot \left[ {\cos \sqrt {\text{x}} \frac{{\text{d}}}{{{\text{dx}}}}\sqrt {\text{x}} } \right]}}{{{{\left( {\sin \sqrt {\text{x}} } \right)}^2}}} \\ \Rightarrow \frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\sin \sqrt {\text{x}} \cdot \frac{{\cos {\text{x}}}}{{2\sqrt {\sin {\text{x}}} }} - \sqrt {\sin {\text{x}}} \cdot \frac{{\cos \sqrt {\text{x}} }}{{2\sqrt {\text{x}} }}}}{{{{\left( {\sin \sqrt {\text{x}} } \right)}^2}}} \\ \Rightarrow \frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\frac{{\sqrt {\text{x}} \cdot \sin \sqrt {\text{x}} \cdot \cos {\text{x}} - \sin {\text{x}} \cdot \cos \sqrt {\text{x}} }}{{2\sqrt {\sin {\text{x}}} \cdot \sqrt {\text{x}} }}}}{{{{\left( {\sin \sqrt {\text{x}} } \right)}^2}}} \\ \Rightarrow \frac{{{\text{dy}}}}{{{\text{dx}}}} = \frac{{\sqrt {\text{x}} \cdot \sin \sqrt {\text{x}} \cdot \cos {\text{x}} - \sin {\text{x}} \cdot \cos \sqrt {\text{x}} }}{{2\sqrt {\sin {\text{x}}} \cdot \sqrt {\text{x}} \cdot {{\sin }^2}\sqrt {\text{x}} }} \\ \end{gathered}