Calculus Examples

$\frac{2}{3} x y^{2} + 2 x - \frac{1}{3} x y^{2} + \frac{2}{3} x^{2} y - 5 x$

Step 1

By the Sum Rule, the derivative of $\frac{2}{3} x y^{2} + 2 x - \frac{1}{3} x y^{2} + \frac{2}{3} x^{2} y - 5 x$ with respect to $x$ is $\frac{d}{d x} [\frac{2}{3} x y^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{1}{3} x y^{2}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$ .

$\frac{d}{d x} [\frac{2}{3} x y^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{1}{3} x y^{2}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 2

Evaluate $\frac{d}{d x} [\frac{2}{3} x y^{2}]$ .

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Step 2.1

Combine $\frac{2}{3}$ and $x$ .

$\frac{d}{d x} [\frac{2 x}{3} y^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{1}{3} x y^{2}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 2.2

Combine $\frac{2 x}{3}$ and $y^{2}$ .

$\frac{d}{d x} [\frac{2 x y^{2}}{3}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{1}{3} x y^{2}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 2.3

Since $\frac{2 y^{2}}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 x y^{2}}{3}$ with respect to $x$ is $\frac{2 y^{2}}{3} \frac{d}{d x} [x]$ .

$\frac{2 y^{2}}{3} \frac{d}{d x} [x] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{1}{3} x y^{2}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 2.4

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

$\frac{2 y^{2}}{3} \cdot 1 + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{1}{3} x y^{2}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 2.5

Multiply $\frac{2 y^{2}}{3}$ by $1$ .

$\frac{2 y^{2}}{3} + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{1}{3} x y^{2}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 3

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

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Step 3.1

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

$\frac{2 y^{2}}{3} + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{1}{3} x y^{2}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 3.2

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

$\frac{2 y^{2}}{3} + 2 \cdot 1 + \frac{d}{d x} [- \frac{1}{3} x y^{2}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 3.3

Multiply $2$ by $1$ .

$\frac{2 y^{2}}{3} + 2 + \frac{d}{d x} [- \frac{1}{3} x y^{2}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 4

Evaluate $\frac{d}{d x} [- \frac{1}{3} x y^{2}]$ .

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Step 4.1

Combine $x$ and $\frac{1}{3}$ .

$\frac{2 y^{2}}{3} + 2 + \frac{d}{d x} [- \frac{x}{3} y^{2}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 4.2

Combine $y^{2}$ and $\frac{x}{3}$ .

$\frac{2 y^{2}}{3} + 2 + \frac{d}{d x} [- \frac{y^{2} x}{3}] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 4.3

Since $- \frac{y^{2}}{3}$ is constant with respect to $x$ , the derivative of $- \frac{y^{2} x}{3}$ with respect to $x$ is $- \frac{y^{2}}{3} \frac{d}{d x} [x]$ .

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 4.4

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

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} \cdot 1 + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 4.5

Multiply $- 1$ by $1$ .

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} + \frac{d}{d x} [\frac{2}{3} x^{2} y] + \frac{d}{d x} [- 5 x]$

Step 5

Evaluate $\frac{d}{d x} [\frac{2}{3} x^{2} y]$ .

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Step 5.1

Combine $\frac{2}{3}$ and $x^{2}$ .

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} + \frac{d}{d x} [\frac{2 x^{2}}{3} y] + \frac{d}{d x} [- 5 x]$

Step 5.2

Combine $\frac{2 x^{2}}{3}$ and $y$ .

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} + \frac{d}{d x} [\frac{2 x^{2} y}{3}] + \frac{d}{d x} [- 5 x]$

Step 5.3

Since $\frac{2 y}{3}$ is constant with respect to $x$ , the derivative of $\frac{2 x^{2} y}{3}$ with respect to $x$ is $\frac{2 y}{3} \frac{d}{d x} [x^{2}]$ .

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} + \frac{2 y}{3} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x]$

Step 5.4

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

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} + \frac{2 y}{3} (2 x) + \frac{d}{d x} [- 5 x]$

Step 5.5

Combine $2$ and $\frac{2 y}{3}$ .

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} + \frac{2 (2 y)}{3} x + \frac{d}{d x} [- 5 x]$

Step 5.6

Multiply $2$ by $2$ .

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} + \frac{4 y}{3} x + \frac{d}{d x} [- 5 x]$

Step 5.7

Combine $\frac{4 y}{3}$ and $x$ .

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} + \frac{4 y x}{3} + \frac{d}{d x} [- 5 x]$

Step 6

Evaluate $\frac{d}{d x} [- 5 x]$ .

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Step 6.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} + \frac{4 y x}{3} - 5 \frac{d}{d x} [x]$

Step 6.2

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

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} + \frac{4 y x}{3} - 5 \cdot 1$

Step 6.3

Multiply $- 5$ by $1$ .

$\frac{2 y^{2}}{3} + 2 - \frac{y^{2}}{3} + \frac{4 y x}{3} - 5$

Step 7

Combine terms.

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Step 7.1

Combine the numerators over the common denominator.

$\frac{2 y^{2} - y^{2}}{3} + 2 + \frac{4 y x}{3} - 5$

Step 7.2

Subtract $y^{2}$ from $2 y^{2}$ .

$\frac{y^{2}}{3} + 2 + \frac{4 y x}{3} - 5$

Step 7.3

Subtract $5$ from $2$ .

$\frac{y^{2}}{3} + \frac{4 y x}{3} - 3$