Calculus Examples

$2 y e^{x} - y \cos (2 x y)$

Step 1

By the Sum Rule, the derivative of $2 y e^{x} - y \cos (2 x y)$ with respect to $y$ is $\frac{d}{d y} [2 y e^{x}] + \frac{d}{d y} [- y \cos (2 x y)]$ .

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

Step 2

Evaluate $\frac{d}{d y} [2 y e^{x}]$ .

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Step 2.1

Since $2 e^{x}$ is constant with respect to $y$ , the derivative of $2 y e^{x}$ with respect to $y$ is $2 e^{x} \frac{d}{d y} [y]$ .

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

Step 2.2

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

$2 e^{x} \cdot 1 + \frac{d}{d y} [- y \cos (2 x y)]$

Step 2.3

Multiply $2$ by $1$ .

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

Step 3

Evaluate $\frac{d}{d y} [- y \cos (2 x y)]$ .

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Step 3.1

Since $- 1$ is constant with respect to $y$ , the derivative of $- y \cos (2 x y)$ with respect to $y$ is $- \frac{d}{d y} [y \cos (2 x y)]$ .

$2 e^{x} - \frac{d}{d y} [y \cos (2 x y)]$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = y$ and $g (y) = \cos (2 x y)$ .

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

Step 3.3

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 $g (y) = 2 x y$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $2 x y$ .

$2 e^{x} - (y (\frac{d}{d u} [\cos (u)] \frac{d}{d y} [2 x y]) + \cos (2 x y) \frac{d}{d y} [y])$

Step 3.3.2

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

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

Step 3.3.3

Replace all occurrences of $u$ with $2 x y$ .

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

Step 3.4

Since $2 x$ is constant with respect to $y$ , the derivative of $2 x y$ with respect to $y$ is $2 x \frac{d}{d y} [y]$ .

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

Step 3.5

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

$2 e^{x} - (y (- \sin (2 x y) (2 x \cdot 1)) + \cos (2 x y) \frac{d}{d y} [y])$

Step 3.6

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

$2 e^{x} - (y (- \sin (2 x y) (2 x \cdot 1)) + \cos (2 x y) \cdot 1)$

Step 3.7

Multiply $2$ by $1$ .

$2 e^{x} - (y (- \sin (2 x y) (2 x)) + \cos (2 x y) \cdot 1)$

Step 3.8

Multiply $2$ by $- 1$ .

$2 e^{x} - (y (- 2 \sin (2 x y) x) + \cos (2 x y) \cdot 1)$

Step 3.9

Multiply $\cos (2 x y)$ by $1$ .

$2 e^{x} - (y (- 2 \sin (2 x y) x) + \cos (2 x y))$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$2 e^{x} - (y (- 2 \sin (2 x y) x)) - \cos (2 x y)$

Step 4.2

Multiply $- 2$ by $- 1$ .

$2 e^{x} + 2 y (\sin (2 x y) x) - \cos (2 x y)$

Step 4.3

Reorder terms.

$2 x y \sin (2 x y) + 2 e^{x} - \cos (2 x y)$