Explained: "Find the first and second derivatives of the..." - Plainmath

Lennie Carroll

Answered question

2021-10-07

Find the first and second derivatives of the function.

y = \cos (x^{2})

Answer & Explanation

komunidadO

Skilled2021-10-08Added 86 answers

Step 1
Given:

y = \cos (x^{2})

Step 2
Differentiating with respect to x:

y = \cos (x^{2})

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

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

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

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

Step 3
Differentiating it again with respect to x :

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

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

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

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