# Second derivatives Find y′′ for the following functions. y=x\sin (x)

Second derivatives Find y′′ for the following functions.
$$\displaystyle{y}={x}{\sin{{\left({x}\right)}}}$$

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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:
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\sin{{\left({x}\right)}}}+{x}{\cos{{\left({x}\right)}}}\right)}$$
$$\displaystyle={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\sin{{\left({x}\right)}}}\right)}{x}{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}{\cos{{\left({x}\right)}}}\right)}$$
$$\displaystyle={x}{\cos{{x}}}{\cos{{x}}}+{\sin{{x}}}{\left[{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{x}{\cos{{x}}}+{x}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\cos{{x}}}\right]}$$
$$\displaystyle={x}{{\cos}^{{{2}}}{x}}+{\sin{{x}}}{\left({\cos{{\left({x}\right)}}}-{x}{\sin{{x}}}\right)}$$
$$\displaystyle={x}{{\cos}^{{{2}}}{x}}+{\sin{{x}}}{\cos{{x}}}-{x}{{\sin}^{{{2}}}{x}}$$
$$\displaystyle={2}{\cos{{\left({x}\right)}}}-{x}{\sin{{\left({x}\right)}}}$$