Calculus Examples

$\frac{7 x^{2} + 5 x^{9}}{4 x^{7}}$

Step 1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{7 x^{2} + 5 x^{9}}{4 x^{7}}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [\frac{7 x^{2} + 5 x^{9}}{x^{7}}]$ .

$\frac{1}{4} \frac{d}{d x} [\frac{7 x^{2} + 5 x^{9}}{x^{7}}]$

Step 2

Factor $x^{2}$ out of $7 x^{2} + 5 x^{9}$ .

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Step 2.1

Factor $x^{2}$ out of $7 x^{2}$ .

$\frac{1}{4} \frac{d}{d x} [\frac{x^{2} \cdot 7 + 5 x^{9}}{x^{7}}]$

Step 2.2

Factor $x^{2}$ out of $5 x^{9}$ .

$\frac{1}{4} \frac{d}{d x} [\frac{x^{2} \cdot 7 + x^{2} (5 x^{7})}{x^{7}}]$

Step 2.3

Factor $x^{2}$ out of $x^{2} \cdot 7 + x^{2} (5 x^{7})$ .

$\frac{1}{4} \frac{d}{d x} [\frac{x^{2} (7 + 5 x^{7})}{x^{7}}]$

Step 3

Cancel the common factors.

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Step 3.1

Factor $x^{2}$ out of $x^{7}$ .

$\frac{1}{4} \frac{d}{d x} [\frac{x^{2} (7 + 5 x^{7})}{x^{2} x^{5}}]$

Step 3.2

Cancel the common factor.

$\frac{1}{4} \frac{d}{d x} [\frac{x^{2} (7 + 5 x^{7})}{x^{2} x^{5}}]$

Step 3.3

Rewrite the expression.

$\frac{1}{4} \frac{d}{d x} [\frac{7 + 5 x^{7}}{x^{5}}]$

Step 4

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 7 + 5 x^{7}$ and $g (x) = x^{5}$ .

$\frac{1}{4} \cdot \frac{x^{5} \frac{d}{d x} [7 + 5 x^{7}] - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{{(x^{5})}^{2}}$

Step 5

Differentiate.

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Step 5.1

Multiply the exponents in ${(x^{5})}^{2}$ .

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Step 5.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{4} \cdot \frac{x^{5} \frac{d}{d x} [7 + 5 x^{7}] - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{5 \cdot 2}}$

Step 5.1.2

Multiply $5$ by $2$ .

$\frac{1}{4} \cdot \frac{x^{5} \frac{d}{d x} [7 + 5 x^{7}] - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 5.2

By the Sum Rule, the derivative of $7 + 5 x^{7}$ with respect to $x$ is $\frac{d}{d x} [7] + \frac{d}{d x} [5 x^{7}]$ .

$\frac{1}{4} \cdot \frac{x^{5} (\frac{d}{d x} [7] + \frac{d}{d x} [5 x^{7}]) - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 5.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$\frac{1}{4} \cdot \frac{x^{5} (0 + \frac{d}{d x} [5 x^{7}]) - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 5.4

Add $0$ and $\frac{d}{d x} [5 x^{7}]$ .

$\frac{1}{4} \cdot \frac{x^{5} \frac{d}{d x} [5 x^{7}] - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 5.5

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{7}$ with respect to $x$ is $5 \frac{d}{d x} [x^{7}]$ .

$\frac{1}{4} \cdot \frac{x^{5} (5 \frac{d}{d x} [x^{7}]) - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 5.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$\frac{1}{4} \cdot \frac{x^{5} (5 (7 x^{6})) - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 5.7

Multiply $7$ by $5$ .

$\frac{1}{4} \cdot \frac{x^{5} (35 x^{6}) - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 6

Multiply $x^{5}$ by $x^{6}$ by adding the exponents.

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Step 6.1

Move $x^{6}$ .

$\frac{1}{4} \cdot \frac{x^{6} x^{5} \cdot 35 - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 6.2

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

$\frac{1}{4} \cdot \frac{x^{6 + 5} \cdot 35 - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 6.3

Add $6$ and $5$ .

$\frac{1}{4} \cdot \frac{x^{11} \cdot 35 - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 7

Move $35$ to the left of $x^{11}$ .

$\frac{1}{4} \cdot \frac{35 \cdot x^{11} - (7 + 5 x^{7}) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{1}{4} \cdot \frac{35 x^{11} - (7 + 5 x^{7}) (5 x^{4})}{x^{10}}$

Step 9

Combine fractions.

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Step 9.1

Multiply $5$ by $- 1$ .

$\frac{1}{4} \cdot \frac{35 x^{11} - 5 (7 + 5 x^{7}) x^{4}}{x^{10}}$

Step 9.2

Multiply $\frac{1}{4}$ by $\frac{35 x^{11} - 5 (7 + 5 x^{7}) x^{4}}{x^{10}}$ .

$\frac{35 x^{11} - 5 (7 + 5 x^{7}) x^{4}}{4 x^{10}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{35 x^{11} + (- 5 \cdot 7 - 5 (5 x^{7})) x^{4}}{4 x^{10}}$

Step 10.2

Apply the distributive property.

$\frac{35 x^{11} - 5 \cdot 7 x^{4} - 5 (5 x^{7}) x^{4}}{4 x^{10}}$

Step 10.3

Simplify the numerator.

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Step 10.3.1

Simplify each term.

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Step 10.3.1.1

Multiply $- 5$ by $7$ .

$\frac{35 x^{11} - 35 x^{4} - 5 (5 x^{7}) x^{4}}{4 x^{10}}$

Step 10.3.1.2

Multiply $x^{7}$ by $x^{4}$ by adding the exponents.

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Step 10.3.1.2.1

Move $x^{4}$ .

$\frac{35 x^{11} - 35 x^{4} - 5 (5 (x^{4} x^{7}))}{4 x^{10}}$

Step 10.3.1.2.2

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

$\frac{35 x^{11} - 35 x^{4} - 5 (5 x^{4 + 7})}{4 x^{10}}$

Step 10.3.1.2.3

Add $4$ and $7$ .

$\frac{35 x^{11} - 35 x^{4} - 5 (5 x^{11})}{4 x^{10}}$

Step 10.3.1.3

Multiply $5$ by $- 5$ .

$\frac{35 x^{11} - 35 x^{4} - 25 x^{11}}{4 x^{10}}$

Step 10.3.2

Subtract $25 x^{11}$ from $35 x^{11}$ .

$\frac{10 x^{11} - 35 x^{4}}{4 x^{10}}$

Step 10.4

Factor $5 x^{4}$ out of $10 x^{11} - 35 x^{4}$ .

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Step 10.4.1

Factor $5 x^{4}$ out of $10 x^{11}$ .

$\frac{5 x^{4} (2 x^{7}) - 35 x^{4}}{4 x^{10}}$

Step 10.4.2

Factor $5 x^{4}$ out of $- 35 x^{4}$ .

$\frac{5 x^{4} (2 x^{7}) + 5 x^{4} (- 7)}{4 x^{10}}$

Step 10.4.3

Factor $5 x^{4}$ out of $5 x^{4} (2 x^{7}) + 5 x^{4} (- 7)$ .

$\frac{5 x^{4} (2 x^{7} - 7)}{4 x^{10}}$

Step 10.5

Cancel the common factor of $x^{4}$ and $x^{10}$ .

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Step 10.5.1

Factor $x^{4}$ out of $5 x^{4} (2 x^{7} - 7)$ .

$\frac{x^{4} (5 (2 x^{7} - 7))}{4 x^{10}}$

Step 10.5.2

Cancel the common factors.

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Step 10.5.2.1

Factor $x^{4}$ out of $4 x^{10}$ .

$\frac{x^{4} (5 (2 x^{7} - 7))}{x^{4} (4 x^{6})}$

Step 10.5.2.2

Cancel the common factor.

$\frac{x^{4} (5 (2 x^{7} - 7))}{x^{4} (4 x^{6})}$

Step 10.5.2.3

Rewrite the expression.

$\frac{5 (2 x^{7} - 7)}{4 x^{6}}$