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

Brittney Lord 2021-11-08 Answered
Second derivatives Find y′′ for the following functions.
\(\displaystyle{y}={x}{\sin{{\left({x}\right)}}}\)

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Expert Answer

lamanocornudaW
Answered 2021-11-09 Author has 12235 answers
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:
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\sin{{\left({x}\right)}}}+{x}{\cos{{\left({x}\right)}}}\right)}\)
\(\displaystyle={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({\sin{{\left({x}\right)}}}\right)}{x}{\cos{{\left({x}\right)}}}+{\sin{{\left({x}\right)}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}{\cos{{\left({x}\right)}}}\right)}\)
\(\displaystyle={x}{\cos{{x}}}{\cos{{x}}}+{\sin{{x}}}{\left[{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{x}{\cos{{x}}}+{x}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\cos{{x}}}\right]}\)
\(\displaystyle={x}{{\cos}^{{{2}}}{x}}+{\sin{{x}}}{\left({\cos{{\left({x}\right)}}}-{x}{\sin{{x}}}\right)}\)
\(\displaystyle={x}{{\cos}^{{{2}}}{x}}+{\sin{{x}}}{\cos{{x}}}-{x}{{\sin}^{{{2}}}{x}}\)
\(\displaystyle={2}{\cos{{\left({x}\right)}}}-{x}{\sin{{\left({x}\right)}}}\)
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