Step 1

Given that:

\(\displaystyle{p}={e}^{{{x}^{{{2}}}{\sin{{x}}}}}\)

Step 2

Differentiate the given function with respect to x then,

To apply chain rules of derivatives,

\(\displaystyle{p}'={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({e}^{{{x}^{{{2}}}{\sin{{x}}}}}\right)}\)

\(\displaystyle={e}^{{{x}^{{{2}}}{\sin{{x}}}}}{\left[{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{x}^{{{2}}}{\sin{{x}}}\right]}\)

\(\displaystyle={e}^{{{x}^{{{2}}}{\sin{{x}}}}}{\left[{x}^{{{2}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\sin{{x}}}+{\sin{{x}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{x}^{{{2}}}\right]}\)

\(\displaystyle={e}^{{{x}^{{{2}}}{\sin{{x}}}}}{\left[{x}^{{{2}}}{\cos{{x}}}+{\sin{{x}}}\cdot{2}{x}\right]}\)

\(\displaystyle{p}'={e}^{{{x}^{{{2}}}{\sin{{x}}}}}{\left[{x}^{{{2}}}{\cos{{x}}}+{2}{x}{\sin{{x}}}\right]}\)

Given that:

\(\displaystyle{p}={e}^{{{x}^{{{2}}}{\sin{{x}}}}}\)

Step 2

Differentiate the given function with respect to x then,

To apply chain rules of derivatives,

\(\displaystyle{p}'={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({e}^{{{x}^{{{2}}}{\sin{{x}}}}}\right)}\)

\(\displaystyle={e}^{{{x}^{{{2}}}{\sin{{x}}}}}{\left[{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{x}^{{{2}}}{\sin{{x}}}\right]}\)

\(\displaystyle={e}^{{{x}^{{{2}}}{\sin{{x}}}}}{\left[{x}^{{{2}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\sin{{x}}}+{\sin{{x}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{x}^{{{2}}}\right]}\)

\(\displaystyle={e}^{{{x}^{{{2}}}{\sin{{x}}}}}{\left[{x}^{{{2}}}{\cos{{x}}}+{\sin{{x}}}\cdot{2}{x}\right]}\)

\(\displaystyle{p}'={e}^{{{x}^{{{2}}}{\sin{{x}}}}}{\left[{x}^{{{2}}}{\cos{{x}}}+{2}{x}{\sin{{x}}}\right]}\)