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

College
Calculus and Analysis
Differential Calculus
Derivatives

Clifland

Answered question

2021-10-03

Find the first and second derivatives of the function.

y = {\cos x}^{2}

Answer & Explanation

unett

Skilled2021-10-04Added 119 answers

Given information:
The given function is $y = \cos (x^{2})$ .
Chain rule:
$\frac{d f (u)}{d x} = \frac{d f}{d u} \cdot \frac{d u}{d x}$
to determine the first derivative:
Let $f = \cos u, u = x^{2}$
$\frac{d y}{d x} = \frac{d}{d u} (\cos (u)) \cdot \frac{d}{d x} (x^{2})$
$= - \sin u (2 x)$
$= - 2 x \sin (x^{2})$
to determine the second derivative:
$\frac{d^{2} y}{{d x}^{2}} = \frac{d}{d x} (- 2 x \sin (x^{2}))$
$= - 2 x (\cos (x^{2})) \cdot 2 x + \sin (x^{2}) (- 2)$
$= - 4 x^{2} \cos (x^{2}) - 2 \sin (x^{2})$

Jeffrey Jordon

Expert2022-11-14Added 2605 answers

Answer is given below (on video)

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