Calculus Examples

$\cos (x y)$

Step 1

This derivative could not be completed using the product rule. Mathway will use another method.

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $x y$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x y$ .

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

Step 3

Differentiate.

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Step 3.1

Since $y$ is constant with respect to $x$ , the derivative of $x y$ with respect to $x$ is $y \frac{d}{d x} [x]$ .

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

Step 3.2

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

$- \sin (x y) (y \cdot 1)$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $y$ by $1$ .

$- \sin (x y) y$

Step 3.3.2

Reorder the factors of $- \sin (x y) y$ .

$- y \sin (x y)$