Get the solution to "Second derivatives Find d2ydx2 for the following functions...."

gorovogpg

Answered question

2021-12-07

Second derivatives Find

d^{2} \frac{y}{{d x}^{2}}

for the following functions.

y = x {\cos x}^{2}

Hector Roberts

Beginner2021-12-08Added 31 answers

Step 1
Given

y = x {\cos x}^{2}

Step 2
Use product rule of derivative

\frac{d}{d x} (f g) = f \frac{d}{d x} g + g \frac{d}{d x} f

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

Step 3

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

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

= - 2 x^{2} {\sin x}^{2} + {\cos x}^{2}

Step 4
Now for second derivative again use product rule of derivative

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

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

= - 4 x^{3} {\cos x}^{2} - 4 x {\sin x}^{2} - 2 x {\sin x}^{2}

= - 4 x^{3} {\cos x}^{2} - 6 x {\sin x}^{2}

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