Calculus Examples

$\frac{1}{5} {(x y^{2} + 4 y)}^{2}$

Step 1

Rewrite ${(x y^{2} + 4 y)}^{2}$ as $(x y^{2} + 4 y) (x y^{2} + 4 y)$ .

$\frac{d}{d x} [\frac{1}{5} ((x y^{2} + 4 y) (x y^{2} + 4 y))]$

Step 2

Expand $(x y^{2} + 4 y) (x y^{2} + 4 y)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [\frac{1}{5} (x y^{2} (x y^{2} + 4 y) + 4 y (x y^{2} + 4 y))]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [\frac{1}{5} (x y^{2} (x y^{2}) + x y^{2} (4 y) + 4 y (x y^{2} + 4 y))]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [\frac{1}{5} (x y^{2} (x y^{2}) + x y^{2} (4 y) + 4 y (x y^{2}) + 4 y (4 y))]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.1.1

Move $x$ .

$\frac{d}{d x} [\frac{1}{5} (x \cdot x y^{2} y^{2} + x y^{2} (4 y) + 4 y (x y^{2}) + 4 y (4 y))]$

Step 3.1.1.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{2} y^{2} + x y^{2} (4 y) + 4 y (x y^{2}) + 4 y (4 y))]$

Step 3.1.2

Multiply $y^{2}$ by $y^{2}$ by adding the exponents.

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Step 3.1.2.1

Move $y^{2}$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} (y^{2} y^{2}) + x y^{2} (4 y) + 4 y (x y^{2}) + 4 y (4 y))]$

Step 3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{2 + 2} + x y^{2} (4 y) + 4 y (x y^{2}) + 4 y (4 y))]$

Step 3.1.2.3

Add $2$ and $2$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + x y^{2} (4 y) + 4 y (x y^{2}) + 4 y (4 y))]$

Step 3.1.3

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{2} y + 4 y (x y^{2}) + 4 y (4 y))]$

Step 3.1.4

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 3.1.4.1

Move $y$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x (y \cdot y^{2}) + 4 y (x y^{2}) + 4 y (4 y))]$

Step 3.1.4.2

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

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Step 3.1.4.2.1

Raise $y$ to the power of $1$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x (y^{1} y^{2}) + 4 y (x y^{2}) + 4 y (4 y))]$

Step 3.1.4.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{1 + 2} + 4 y (x y^{2}) + 4 y (4 y))]$

Step 3.1.4.3

Add $1$ and $2$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{3} + 4 y (x y^{2}) + 4 y (4 y))]$

Step 3.1.5

Multiply $y$ by $y^{2}$ by adding the exponents.

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Step 3.1.5.1

Move $y^{2}$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{3} + 4 (y^{2} y) x + 4 y (4 y))]$

Step 3.1.5.2

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

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Step 3.1.5.2.1

Raise $y$ to the power of $1$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{3} + 4 (y^{2} y^{1}) x + 4 y (4 y))]$

Step 3.1.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{3} + 4 y^{2 + 1} x + 4 y (4 y))]$

Step 3.1.5.3

Add $2$ and $1$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{3} + 4 y^{3} x + 4 y (4 y))]$

Step 3.1.6

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{3} + 4 y^{3} x + 4 \cdot 4 y \cdot y)]$

Step 3.1.7

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.7.1

Move $y$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{3} + 4 y^{3} x + 4 \cdot 4 (y \cdot y))]$

Step 3.1.7.2

Multiply $y$ by $y$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{3} + 4 y^{3} x + 4 \cdot 4 y^{2})]$

Step 3.1.8

Multiply $4$ by $4$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{3} + 4 y^{3} x + 16 y^{2})]$

Step 3.2

Add $4 x y^{3}$ and $4 y^{3} x$ .

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Step 3.2.1

Move $y^{3}$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 4 x y^{3} + 4 x y^{3} + 16 y^{2})]$

Step 3.2.2

Add $4 x y^{3}$ and $4 x y^{3}$ .

$\frac{d}{d x} [\frac{1}{5} (x^{2} y^{4} + 8 x y^{3} + 16 y^{2})]$

Step 4

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{1}{5} (x^{2} y^{4} + 8 x y^{3} + 16 y^{2})$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [x^{2} y^{4} + 8 x y^{3} + 16 y^{2}]$ .

$\frac{1}{5} \frac{d}{d x} [x^{2} y^{4} + 8 x y^{3} + 16 y^{2}]$

Step 5

By the Sum Rule, the derivative of $x^{2} y^{4} + 8 x y^{3} + 16 y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2} y^{4}] + \frac{d}{d x} [8 x y^{3}] + \frac{d}{d x} [16 y^{2}]$ .

$\frac{1}{5} (\frac{d}{d x} [x^{2} y^{4}] + \frac{d}{d x} [8 x y^{3}] + \frac{d}{d x} [16 y^{2}])$

Step 6

Since $y^{4}$ is constant with respect to $x$ , the derivative of $x^{2} y^{4}$ with respect to $x$ is $y^{4} \frac{d}{d x} [x^{2}]$ .

$\frac{1}{5} (y^{4} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x y^{3}] + \frac{d}{d x} [16 y^{2}])$

Step 7

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

$\frac{1}{5} (y^{4} (2 x) + \frac{d}{d x} [8 x y^{3}] + \frac{d}{d x} [16 y^{2}])$

Step 8

Move $2$ to the left of $y^{4}$ .

$\frac{1}{5} (2 \cdot y^{4} x + \frac{d}{d x} [8 x y^{3}] + \frac{d}{d x} [16 y^{2}])$

Step 9

Since $8 y^{3}$ is constant with respect to $x$ , the derivative of $8 x y^{3}$ with respect to $x$ is $8 y^{3} \frac{d}{d x} [x]$ .

$\frac{1}{5} (2 y^{4} x + 8 y^{3} \frac{d}{d x} [x] + \frac{d}{d x} [16 y^{2}])$

Step 10

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

$\frac{1}{5} (2 y^{4} x + 8 y^{3} \cdot 1 + \frac{d}{d x} [16 y^{2}])$

Step 11

Multiply $8$ by $1$ .

$\frac{1}{5} (2 y^{4} x + 8 y^{3} + \frac{d}{d x} [16 y^{2}])$

Step 12

Since $16 y^{2}$ is constant with respect to $x$ , the derivative of $16 y^{2}$ with respect to $x$ is $0$ .

$\frac{1}{5} (2 y^{4} x + 8 y^{3} + 0)$

Step 13

Add $2 y^{4} x + 8 y^{3}$ and $0$ .

$\frac{1}{5} (2 y^{4} x + 8 y^{3})$

Step 14

Simplify.

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Step 14.1

Apply the distributive property.

$\frac{1}{5} (2 y^{4} x) + \frac{1}{5} (8 y^{3})$

Step 14.2

Combine terms.

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Step 14.2.1

Combine $2$ and $\frac{1}{5}$ .

$\frac{2}{5} (y^{4} x) + \frac{1}{5} (8 y^{3})$

Step 14.2.2

Combine $y^{4}$ and $\frac{2}{5}$ .

$\frac{y^{4} \cdot 2}{5} x + \frac{1}{5} (8 y^{3})$

Step 14.2.3

Combine $\frac{y^{4} \cdot 2}{5}$ and $x$ .

$\frac{y^{4} \cdot 2 x}{5} + \frac{1}{5} (8 y^{3})$

Step 14.2.4

Move $2$ to the left of $y^{4}$ .

$\frac{2 \cdot y^{4} x}{5} + \frac{1}{5} (8 y^{3})$

Step 14.2.5

Combine $8$ and $\frac{1}{5}$ .

$\frac{2 y^{4} x}{5} + \frac{8}{5} y^{3}$

Step 14.2.6

Combine $\frac{8}{5}$ and $y^{3}$ .

$\frac{2 y^{4} x}{5} + \frac{8 y^{3}}{5}$