Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Multiply $y^{2}$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.3

By the Sum Rule, the derivative of $x + y$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [y]$ .

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

Step 3.4

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

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

Step 3.5

Since $y$ is constant with respect to $x$ , the derivative of $y$ with respect to $x$ is $0$ .

$y^{2} - (\cos (x + y) (1 + 0)) + \frac{d}{d x} [y]$

Step 3.6

Add $1$ and $0$ .

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

Step 3.7

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

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

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $y$ is constant with respect to $x$ , the derivative of $y$ with respect to $x$ is $0$ .

$y^{2} - \cos (x + y) + 0$

Step 4.2

Add $y^{2} - \cos (x + y)$ and $0$ .

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