Calculus Examples

$y = \frac{1}{x^{4}} \cdot \sqrt{196 x^{8} + 420 x^{3} + \frac{225}{x^{2}}}$

Step 1

Rewrite $\frac{d}{d x} [\frac{1}{x^{4}} \cdot \sqrt{196 x^{8} + 420 x^{3} + \frac{225}{x^{2}}}]$ as $\frac{d}{d x} [\frac{1}{x^{4}} \cdot (14 x^{4} + \frac{15}{x})]$ .

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $196 x^{8}$ as ${(14 x^{4})}^{2}$ .

$\frac{d}{d x} [\frac{1}{x^{4}} \cdot \sqrt{{(14 x^{4})}^{2} + 420 x^{3} + \frac{225}{x^{2}}}]$

Step 1.1.2

Rewrite $225$ as $15^{2}$ .

$\frac{d}{d x} [\frac{1}{x^{4}} \cdot \sqrt{{(14 x^{4})}^{2} + 420 x^{3} + \frac{15^{2}}{x^{2}}}]$

Step 1.1.3

Rewrite $\frac{15^{2}}{x^{2}}$ as ${(\frac{15}{x})}^{2}$ .

$\frac{d}{d x} [\frac{1}{x^{4}} \cdot \sqrt{{(14 x^{4})}^{2} + 420 x^{3} + {(\frac{15}{x})}^{2}}]$

Step 1.1.4

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$420 x^{3} = 2 \cdot (14 x^{4}) \cdot \frac{15}{x}$

Step 1.1.5

Rewrite the polynomial.

$\frac{d}{d x} [\frac{1}{x^{4}} \cdot \sqrt{{(14 x^{4})}^{2} + 2 \cdot (14 x^{4}) \cdot \frac{15}{x} + {(\frac{15}{x})}^{2}}]$

Step 1.1.6

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 14 x^{4}$ and $b = \frac{15}{x}$ .

$\frac{d}{d x} [\frac{1}{x^{4}} \cdot \sqrt{{(14 x^{4} + \frac{15}{x})}^{2}}]$

Step 1.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{d}{d x} [\frac{1}{x^{4}} \cdot (14 x^{4} + \frac{15}{x})]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \frac{1}{x^{4}}$ and $g (x) = 14 x^{4} + \frac{15}{x}$ .

$\frac{1}{x^{4}} \frac{d}{d x} [14 x^{4} + \frac{15}{x}] + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $14 x^{4} + \frac{15}{x}$ with respect to $x$ is $\frac{d}{d x} [14 x^{4}] + \frac{d}{d x} [\frac{15}{x}]$ .

$\frac{1}{x^{4}} (\frac{d}{d x} [14 x^{4}] + \frac{d}{d x} [\frac{15}{x}]) + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3.2

Since $14$ is constant with respect to $x$ , the derivative of $14 x^{4}$ with respect to $x$ is $14 \frac{d}{d x} [x^{4}]$ .

$\frac{1}{x^{4}} (14 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{15}{x}]) + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{x^{4}} (14 (4 x^{3}) + \frac{d}{d x} [\frac{15}{x}]) + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3.4

Multiply $4$ by $14$ .

$\frac{1}{x^{4}} (56 x^{3} + \frac{d}{d x} [\frac{15}{x}]) + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3.5

Since $15$ is constant with respect to $x$ , the derivative of $\frac{15}{x}$ with respect to $x$ is $15 \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{1}{x^{4}} (56 x^{3} + 15 \frac{d}{d x} [\frac{1}{x}]) + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3.6

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$\frac{1}{x^{4}} (56 x^{3} + 15 \frac{d}{d x} [x^{- 1}]) + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3.7

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

$\frac{1}{x^{4}} (56 x^{3} + 15 (- x^{- 2})) + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3.8

Simplify the expression.

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Step 3.8.1

Multiply $- 1$ by $15$ .

$\frac{1}{x^{4}} (56 x^{3} - 15 x^{- 2}) + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 3.8.2

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$\frac{1}{x^{4}} (56 x^{3} - 15 x^{- 2}) + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [{(x^{4})}^{- 1}]$

Step 3.8.3

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 3.8.3.1

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

$\frac{1}{x^{4}} (56 x^{3} - 15 x^{- 2}) + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [x^{4 \cdot - 1}]$

Step 3.8.3.2

Multiply $4$ by $- 1$ .

$\frac{1}{x^{4}} (56 x^{3} - 15 x^{- 2}) + (14 x^{4} + \frac{15}{x}) \frac{d}{d x} [x^{- 4}]$

Step 3.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = - 4$ .

$\frac{1}{x^{4}} (56 x^{3} - 15 x^{- 2}) + (14 x^{4} + \frac{15}{x}) (- 4 x^{- 5})$

Step 4

Simplify.

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{4}} (56 x^{3} - 15 \frac{1}{x^{2}}) + (14 x^{4} + \frac{15}{x}) (- 4 x^{- 5})$

Step 4.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{x^{4}} (56 x^{3} - 15 \frac{1}{x^{2}}) + (14 x^{4} + \frac{15}{x}) (- 4 \frac{1}{x^{5}})$

Step 4.3

Apply the distributive property.

$\frac{1}{x^{4}} (56 x^{3}) + \frac{1}{x^{4}} (- 15 \frac{1}{x^{2}}) + (14 x^{4} + \frac{15}{x}) (- 4 \frac{1}{x^{5}})$

Step 4.4

Apply the distributive property.

$\frac{1}{x^{4}} (56 x^{3}) + \frac{1}{x^{4}} (- 15 \frac{1}{x^{2}}) + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5

Combine terms.

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Step 4.5.1

Combine $56$ and $\frac{1}{x^{4}}$ .

$\frac{56}{x^{4}} x^{3} + \frac{1}{x^{4}} (- 15 \frac{1}{x^{2}}) + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.2

Combine $\frac{56}{x^{4}}$ and $x^{3}$ .

$\frac{56 x^{3}}{x^{4}} + \frac{1}{x^{4}} (- 15 \frac{1}{x^{2}}) + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.3

Cancel the common factor of $x^{3}$ and $x^{4}$ .

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Step 4.5.3.1

Factor $x^{3}$ out of $56 x^{3}$ .

$\frac{x^{3} \cdot 56}{x^{4}} + \frac{1}{x^{4}} (- 15 \frac{1}{x^{2}}) + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.3.2

Cancel the common factors.

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Step 4.5.3.2.1

Factor $x^{3}$ out of $x^{4}$ .

$\frac{x^{3} \cdot 56}{x^{3} x} + \frac{1}{x^{4}} (- 15 \frac{1}{x^{2}}) + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.3.2.2

Cancel the common factor.

$\frac{x^{3} \cdot 56}{x^{3} x} + \frac{1}{x^{4}} (- 15 \frac{1}{x^{2}}) + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.3.2.3

Rewrite the expression.

$\frac{56}{x} + \frac{1}{x^{4}} (- 15 \frac{1}{x^{2}}) + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.4

Combine $- 15$ and $\frac{1}{x^{2}}$ .

$\frac{56}{x} + \frac{1}{x^{4}} \cdot \frac{- 15}{x^{2}} + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.5

Move the negative in front of the fraction.

$\frac{56}{x} + \frac{1}{x^{4}} (- \frac{15}{x^{2}}) + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.6

Multiply $\frac{1}{x^{4}}$ by $\frac{15}{x^{2}}$ .

$\frac{56}{x} - \frac{15}{x^{4} x^{2}} + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.7

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 4.5.7.1

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

$\frac{56}{x} - \frac{15}{x^{4 + 2}} + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.7.2

Add $4$ and $2$ .

$\frac{56}{x} - \frac{15}{x^{6}} + 14 x^{4} (- 4 \frac{1}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.8

Combine $- 4$ and $\frac{1}{x^{5}}$ .

$\frac{56}{x} - \frac{15}{x^{6}} + 14 x^{4} \frac{- 4}{x^{5}} + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.9

Move the negative in front of the fraction.

$\frac{56}{x} - \frac{15}{x^{6}} + 14 x^{4} (- \frac{4}{x^{5}}) + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.10

Multiply $- 1$ by $14$ .

$\frac{56}{x} - \frac{15}{x^{6}} - 14 x^{4} \frac{4}{x^{5}} + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.11

Combine $- 14$ and $\frac{4}{x^{5}}$ .

$\frac{56}{x} - \frac{15}{x^{6}} + x^{4} \frac{- 14 \cdot 4}{x^{5}} + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.12

Multiply $- 14$ by $4$ .

$\frac{56}{x} - \frac{15}{x^{6}} + x^{4} \frac{- 56}{x^{5}} + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.13

Combine $x^{4}$ and $\frac{- 56}{x^{5}}$ .

$\frac{56}{x} - \frac{15}{x^{6}} + \frac{x^{4} \cdot - 56}{x^{5}} + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.14

Move $- 56$ to the left of $x^{4}$ .

$\frac{56}{x} - \frac{15}{x^{6}} + \frac{- 56 \cdot x^{4}}{x^{5}} + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.15

Cancel the common factor of $x^{4}$ and $x^{5}$ .

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Step 4.5.15.1

Factor $x^{4}$ out of $- 56 x^{4}$ .

$\frac{56}{x} - \frac{15}{x^{6}} + \frac{x^{4} \cdot - 56}{x^{5}} + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.15.2

Cancel the common factors.

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Step 4.5.15.2.1

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

$\frac{56}{x} - \frac{15}{x^{6}} + \frac{x^{4} \cdot - 56}{x^{4} x} + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.15.2.2

Cancel the common factor.

$\frac{56}{x} - \frac{15}{x^{6}} + \frac{x^{4} \cdot - 56}{x^{4} x} + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.15.2.3

Rewrite the expression.

$\frac{56}{x} - \frac{15}{x^{6}} + \frac{- 56}{x} + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.16

Move the negative in front of the fraction.

$\frac{56}{x} - \frac{15}{x^{6}} - \frac{56}{x} + \frac{15}{x} (- 4 \frac{1}{x^{5}})$

Step 4.5.17

Combine $- 4$ and $\frac{1}{x^{5}}$ .

$\frac{56}{x} - \frac{15}{x^{6}} - \frac{56}{x} + \frac{15}{x} \cdot \frac{- 4}{x^{5}}$

Step 4.5.18

Move the negative in front of the fraction.

$\frac{56}{x} - \frac{15}{x^{6}} - \frac{56}{x} + \frac{15}{x} (- \frac{4}{x^{5}})$

Step 4.5.19

Multiply $\frac{15}{x}$ by $\frac{4}{x^{5}}$ .

$\frac{56}{x} - \frac{15}{x^{6}} - \frac{56}{x} - \frac{15 \cdot 4}{x \cdot x^{5}}$

Step 4.5.20

Multiply $15$ by $4$ .

$\frac{56}{x} - \frac{15}{x^{6}} - \frac{56}{x} - \frac{60}{x \cdot x^{5}}$

Step 4.5.21

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

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Step 4.5.21.1

Multiply $x$ by $x^{5}$ .

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Step 4.5.21.1.1

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

$\frac{56}{x} - \frac{15}{x^{6}} - \frac{56}{x} - \frac{60}{x^{1} x^{5}}$

Step 4.5.21.1.2

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

$\frac{56}{x} - \frac{15}{x^{6}} - \frac{56}{x} - \frac{60}{x^{1 + 5}}$

Step 4.5.21.2

Add $1$ and $5$ .

$\frac{56}{x} - \frac{15}{x^{6}} - \frac{56}{x} - \frac{60}{x^{6}}$

Step 4.5.22

Subtract $\frac{56}{x}$ from $\frac{56}{x}$ .

$- \frac{15}{x^{6}} + 0 - \frac{60}{x^{6}}$

Step 4.5.23

Add $- \frac{15}{x^{6}}$ and $0$ .

$- \frac{15}{x^{6}} - \frac{60}{x^{6}}$

Step 4.5.24

Combine the numerators over the common denominator.

$\frac{- 15 - 60}{x^{6}}$

Step 4.5.25

Subtract $60$ from $- 15$ .

$\frac{- 75}{x^{6}}$

Step 4.5.26

Move the negative in front of the fraction.

$- \frac{75}{x^{6}}$