Key to "Second derivatives Find y″ for the following functions...."

Albarellak

Answered question

2021-05-02

Second derivatives Find y″ for the following functions.

y = x \sin x

avortarF

Skilled2021-05-03Added 113 answers

Step 1. Given that
Second derivatives Find y″ for the following functions.

y = x \sin x

Step 2: Formula Used
Product Rule of Differentiation
(u.v)=u'v+v'u
Step 3: Finding the double derivative
We have,

y = x \sin x

differentiating both sides w.r.t x we obtain,

\frac{d y}{d x} = {(x)}^{'} . \sin x + {(\sin x)}^{'} . x

\frac{d y}{d x} = \sin x + x \cos x

Again differentiating the above equation both sides w.r.t x we obtain

\frac{d}{d x} (\frac{d y}{d x}) = \frac{d}{d x} (\sin x + x \cos x)

\frac{d^{2} y}{{d x}^{2}} = \cos x + ({(x)}^{'} \cos x + {(\cos x)}^{'} x)

\frac{d^{2} y}{{d x}^{2}} = \cos x + (\cos x - x \sin x)

\frac{d^{2} y}{{d x}^{2}} = 2 \cos x - x \sin x

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