Step 1

Here given function,

\(\displaystyle{y}={x}{\sin{{\left({x}\right)}}}\)

and we have to find the second derivatives of the given function.

concept used: product rules of derivation

Rule is.

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{f{{\left({x}\right)}}}\cdot{g{{\left({x}\right)}}}\right]}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{f{{\left({x}\right)}}}\right]}\cdot{g{{\left({x}\right)}}}+{f{{\left({x}\right)}}}\cdot{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{g{{\left({x}\right)}}}\right]}\)

Here,

f(x)=x

\(\displaystyle{g{{\left({x}\right)}}}={\sin{{\left({x}\right)}}}\)

we find first derivative of given function and then find second derivative as follows.

Step 2

Rewrite the function:

\(\displaystyle{\frac{{{d}^{{{2}}}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}{\left({x}{\sin{{\left({x}\right)}}}\right)}\)

\(\displaystyle={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}{\sin{{\left({x}\right)}}}\right)}\)

\(\displaystyle={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}\right)}{\sin{{\left({x}\right)}}}+{x}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\sin{{\left({x}\right)}}}\)

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}{\sin{{\left({x}\right)}}}\right)}={\sin{{\left({x}\right)}}}+{x}{\cos{{\left({x}\right)}}}\)

this is first derivative now differentiate it again to get second derivative:

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\sin{{\left({x}\right)}}}+{x}{\cos{{\left({x}\right)}}}\right)}\)

\(\displaystyle={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\sin{{\left({x}\right)}}}\right)}{x}{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}{\cos{{\left({x}\right)}}}\right)}\)

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

\(\displaystyle={x}{{\cos}^{{{2}}}{x}}+{\sin{{x}}}{\left({\cos{{\left({x}\right)}}}-{x}{\sin{{x}}}\right)}\)

\(\displaystyle={x}{{\cos}^{{{2}}}{x}}+{\sin{{x}}}{\cos{{x}}}-{x}{{\sin}^{{{2}}}{x}}\)

\(\displaystyle={2}{\cos{{\left({x}\right)}}}-{x}{\sin{{\left({x}\right)}}}\)

Here given function,

\(\displaystyle{y}={x}{\sin{{\left({x}\right)}}}\)

and we have to find the second derivatives of the given function.

concept used: product rules of derivation

Rule is.

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{f{{\left({x}\right)}}}\cdot{g{{\left({x}\right)}}}\right]}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{f{{\left({x}\right)}}}\right]}\cdot{g{{\left({x}\right)}}}+{f{{\left({x}\right)}}}\cdot{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{g{{\left({x}\right)}}}\right]}\)

Here,

f(x)=x

\(\displaystyle{g{{\left({x}\right)}}}={\sin{{\left({x}\right)}}}\)

we find first derivative of given function and then find second derivative as follows.

Step 2

Rewrite the function:

\(\displaystyle{\frac{{{d}^{{{2}}}}}{{{\left.{d}{x}\right.}^{{{2}}}}}}{\left({x}{\sin{{\left({x}\right)}}}\right)}\)

\(\displaystyle={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}{\sin{{\left({x}\right)}}}\right)}\)

\(\displaystyle={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}\right)}{\sin{{\left({x}\right)}}}+{x}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\sin{{\left({x}\right)}}}\)

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}{\sin{{\left({x}\right)}}}\right)}={\sin{{\left({x}\right)}}}+{x}{\cos{{\left({x}\right)}}}\)

this is first derivative now differentiate it again to get second derivative:

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\sin{{\left({x}\right)}}}+{x}{\cos{{\left({x}\right)}}}\right)}\)

\(\displaystyle={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\sin{{\left({x}\right)}}}\right)}{x}{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}{\cos{{\left({x}\right)}}}\right)}\)

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

\(\displaystyle={x}{{\cos}^{{{2}}}{x}}+{\sin{{x}}}{\left({\cos{{\left({x}\right)}}}-{x}{\sin{{x}}}\right)}\)

\(\displaystyle={x}{{\cos}^{{{2}}}{x}}+{\sin{{x}}}{\cos{{x}}}-{x}{{\sin}^{{{2}}}{x}}\)

\(\displaystyle={2}{\cos{{\left({x}\right)}}}-{x}{\sin{{\left({x}\right)}}}\)