Calculus Examples

$\cos (y^{2})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = \cos (y)$ and $g (y) = y^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $y^{2}$ .

$\frac{d}{d u} [\cos (u)] \frac{d}{d y} [y^{2}]$

Step 1.2

The derivative of $\cos (u)$ with respect to $u$ is $- \sin (u)$ .

$- \sin (u) \frac{d}{d y} [y^{2}]$

Step 1.3

Replace all occurrences of $u$ with $y^{2}$ .

$- \sin (y^{2}) \frac{d}{d y} [y^{2}]$

Step 2

Differentiate using the Power Rule.

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Step 2.1

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 2$ .

$- \sin (y^{2}) (2 y)$

Step 2.2

Simplify the expression.

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Step 2.2.1

Multiply $2$ by $- 1$ .

$- 2 \sin (y^{2}) y$

Step 2.2.2

Reorder the factors of $- 2 \sin (y^{2}) y$ .

$- 2 y \sin (y^{2})$