Answered: "Second derivatives Find y″ for the following functions...." - Plainmath

hunterofdeath63

Answered question

2021-12-07

Second derivatives Find y″ for the following functions.

y = x^{2} \cos x

Answer & Explanation

Jacob Homer

Beginner2021-12-08Added 41 answers

Step 1
Consider the given:

y = x^{2} \cos x

Differentiate w.r.t. "x",

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

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

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

y^{'} = 2 x \cos (x) - x^{2} \sin (x)

Step 2
Again differentiate w.r.t. “x” for second derivative:

y^{'} = 2 x \cos (x) - x^{2} \sin (x)

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

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

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

y^{″} = 2 \cos x - 2 x \sin x - 2 x \sin (x) - x^{2} \cos x

y^{″} = 2 \cos x - 4 x \sin x - x^{2} \cos x

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