Calculus Examples

$y = \frac{3 x^{5} - 7 x^{2 - 4}}{x^{2}}$

Step 1

Find the first derivative.

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Step 1.1

Subtract $4$ from $2$ .

$\frac{d}{d x} [\frac{3 x^{5} - 7 x^{- 2}}{x^{2}}]$

Step 1.2

Factor $x^{- 2}$ out of $3 x^{5} - 7 x^{- 2}$ .

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Step 1.2.1

Factor $x^{- 2}$ out of $3 x^{5}$ .

$\frac{d}{d x} [\frac{x^{- 2} (3 x^{7}) - 7 x^{- 2}}{x^{2}}]$

Step 1.2.2

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

$\frac{d}{d x} [\frac{x^{- 2} (3 x^{7}) + x^{- 2} \cdot - 7}{x^{2}}]$

Step 1.2.3

Factor $x^{- 2}$ out of $x^{- 2} (3 x^{7}) + x^{- 2} \cdot - 7$ .

$\frac{d}{d x} [\frac{x^{- 2} (3 x^{7} - 7)}{x^{2}}]$

Step 1.3

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{d}{d x} [\frac{3 x^{7} - 7}{x^{2} x^{2}}]$

Step 1.4

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

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Step 1.4.1

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

$\frac{d}{d x} [\frac{3 x^{7} - 7}{x^{2 + 2}}]$

Step 1.4.2

Add $2$ and $2$ .

$\frac{d}{d x} [\frac{3 x^{7} - 7}{x^{4}}]$

Step 1.5

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 $f (x) = 3 x^{7} - 7$ and $g (x) = x^{4}$ .

$\frac{x^{4} \frac{d}{d x} [3 x^{7} - 7] - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{{(x^{4})}^{2}}$

Step 1.6

Differentiate.

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Step 1.6.1

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 1.6.1.1

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

$\frac{x^{4} \frac{d}{d x} [3 x^{7} - 7] - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{4 \cdot 2}}$

Step 1.6.1.2

Multiply $4$ by $2$ .

$\frac{x^{4} \frac{d}{d x} [3 x^{7} - 7] - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 1.6.2

By the Sum Rule, the derivative of $3 x^{7} - 7$ with respect to $x$ is $\frac{d}{d x} [3 x^{7}] + \frac{d}{d x} [- 7]$ .

$\frac{x^{4} (\frac{d}{d x} [3 x^{7}] + \frac{d}{d x} [- 7]) - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 1.6.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{7}$ with respect to $x$ is $3 \frac{d}{d x} [x^{7}]$ .

$\frac{x^{4} (3 \frac{d}{d x} [x^{7}] + \frac{d}{d x} [- 7]) - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 1.6.4

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

$\frac{x^{4} (3 (7 x^{6}) + \frac{d}{d x} [- 7]) - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 1.6.5

Multiply $7$ by $3$ .

$\frac{x^{4} (21 x^{6} + \frac{d}{d x} [- 7]) - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 1.6.6

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

$\frac{x^{4} (21 x^{6} + 0) - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 1.6.7

Add $21 x^{6}$ and $0$ .

$\frac{x^{4} (21 x^{6}) - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 1.7

Multiply $x^{4}$ by $x^{6}$ by adding the exponents.

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Step 1.7.1

Move $x^{6}$ .

$\frac{x^{6} x^{4} \cdot 21 - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 1.7.2

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

$\frac{x^{6 + 4} \cdot 21 - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 1.7.3

Add $6$ and $4$ .

$\frac{x^{10} \cdot 21 - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 1.8

Move $21$ to the left of $x^{10}$ .

$\frac{21 \cdot x^{10} - (3 x^{7} - 7) \frac{d}{d x} [x^{4}]}{x^{8}}$

Step 1.9

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

$\frac{21 x^{10} - (3 x^{7} - 7) (4 x^{3})}{x^{8}}$

Step 1.10

Simplify with factoring out.

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Step 1.10.1

Multiply $4$ by $- 1$ .

$\frac{21 x^{10} - 4 (3 x^{7} - 7) x^{3}}{x^{8}}$

Step 1.10.2

Factor $x^{3}$ out of $21 x^{10} - 4 (3 x^{7} - 7) x^{3}$ .

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Step 1.10.2.1

Factor $x^{3}$ out of $21 x^{10}$ .

$\frac{x^{3} (21 x^{7}) - 4 (3 x^{7} - 7) x^{3}}{x^{8}}$

Step 1.10.2.2

Factor $x^{3}$ out of $- 4 (3 x^{7} - 7) x^{3}$ .

$\frac{x^{3} (21 x^{7}) + x^{3} (- 4 (3 x^{7} - 7))}{x^{8}}$

Step 1.10.2.3

Factor $x^{3}$ out of $x^{3} (21 x^{7}) + x^{3} (- 4 (3 x^{7} - 7))$ .

$\frac{x^{3} (21 x^{7} - 4 (3 x^{7} - 7))}{x^{8}}$

Step 1.11

Cancel the common factors.

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Step 1.11.1

Factor $x^{3}$ out of $x^{8}$ .

$\frac{x^{3} (21 x^{7} - 4 (3 x^{7} - 7))}{x^{3} x^{5}}$

Step 1.11.2

Cancel the common factor.

$\frac{x^{3} (21 x^{7} - 4 (3 x^{7} - 7))}{x^{3} x^{5}}$

Step 1.11.3

Rewrite the expression.

$\frac{21 x^{7} - 4 (3 x^{7} - 7)}{x^{5}}$

Step 1.12

Simplify.

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Step 1.12.1

Apply the distributive property.

$\frac{21 x^{7} - 4 (3 x^{7}) - 4 \cdot - 7}{x^{5}}$

Step 1.12.2

Simplify the numerator.

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Step 1.12.2.1

Simplify each term.

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Step 1.12.2.1.1

Multiply $3$ by $- 4$ .

$\frac{21 x^{7} - 12 x^{7} - 4 \cdot - 7}{x^{5}}$

Step 1.12.2.1.2

Multiply $- 4$ by $- 7$ .

$\frac{21 x^{7} - 12 x^{7} + 28}{x^{5}}$

Step 1.12.2.2

Subtract $12 x^{7}$ from $21 x^{7}$ .

$f' (x) = \frac{9 x^{7} + 28}{x^{5}}$

Step 2

Find the second derivative.

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Step 2.1

$\frac{x^{5} \frac{d}{d x} [9 x^{7} + 28] - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{{(x^{5})}^{2}}$

Step 2.2

Differentiate.

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Step 2.2.1

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

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Step 2.2.1.1

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

$\frac{x^{5} \frac{d}{d x} [9 x^{7} + 28] - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{5 \cdot 2}}$

Step 2.2.1.2

Multiply $5$ by $2$ .

$\frac{x^{5} \frac{d}{d x} [9 x^{7} + 28] - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.2.2

By the Sum Rule, the derivative of $9 x^{7} + 28$ with respect to $x$ is $\frac{d}{d x} [9 x^{7}] + \frac{d}{d x} [28]$ .

$\frac{x^{5} (\frac{d}{d x} [9 x^{7}] + \frac{d}{d x} [28]) - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.2.3

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

$\frac{x^{5} (9 \frac{d}{d x} [x^{7}] + \frac{d}{d x} [28]) - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.2.4

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

$\frac{x^{5} (9 (7 x^{6}) + \frac{d}{d x} [28]) - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.2.5

Multiply $7$ by $9$ .

$\frac{x^{5} (63 x^{6} + \frac{d}{d x} [28]) - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.2.6

Since $28$ is constant with respect to $x$ , the derivative of $28$ with respect to $x$ is $0$ .

$\frac{x^{5} (63 x^{6} + 0) - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.2.7

Add $63 x^{6}$ and $0$ .

$\frac{x^{5} (63 x^{6}) - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.3

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

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Step 2.3.1

Move $x^{6}$ .

$\frac{x^{6} x^{5} \cdot 63 - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.3.2

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

$\frac{x^{6 + 5} \cdot 63 - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.3.3

Add $6$ and $5$ .

$\frac{x^{11} \cdot 63 - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.4

Move $63$ to the left of $x^{11}$ .

$\frac{63 \cdot x^{11} - (9 x^{7} + 28) \frac{d}{d x} [x^{5}]}{x^{10}}$

Step 2.5

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

$\frac{63 x^{11} - (9 x^{7} + 28) (5 x^{4})}{x^{10}}$

Step 2.6

Simplify with factoring out.

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Step 2.6.1

Multiply $5$ by $- 1$ .

$\frac{63 x^{11} - 5 (9 x^{7} + 28) x^{4}}{x^{10}}$

Step 2.6.2

Factor $x^{4}$ out of $63 x^{11} - 5 (9 x^{7} + 28) x^{4}$ .

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Step 2.6.2.1

Factor $x^{4}$ out of $63 x^{11}$ .

$\frac{x^{4} (63 x^{7}) - 5 (9 x^{7} + 28) x^{4}}{x^{10}}$

Step 2.6.2.2

Factor $x^{4}$ out of $- 5 (9 x^{7} + 28) x^{4}$ .

$\frac{x^{4} (63 x^{7}) + x^{4} (- 5 (9 x^{7} + 28))}{x^{10}}$

Step 2.6.2.3

Factor $x^{4}$ out of $x^{4} (63 x^{7}) + x^{4} (- 5 (9 x^{7} + 28))$ .

$\frac{x^{4} (63 x^{7} - 5 (9 x^{7} + 28))}{x^{10}}$

Step 2.7

Cancel the common factors.

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Step 2.7.1

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

$\frac{x^{4} (63 x^{7} - 5 (9 x^{7} + 28))}{x^{4} x^{6}}$

Step 2.7.2

Cancel the common factor.

$\frac{x^{4} (63 x^{7} - 5 (9 x^{7} + 28))}{x^{4} x^{6}}$

Step 2.7.3

Rewrite the expression.

$\frac{63 x^{7} - 5 (9 x^{7} + 28)}{x^{6}}$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$\frac{63 x^{7} - 5 (9 x^{7}) - 5 \cdot 28}{x^{6}}$

Step 2.8.2

Simplify the numerator.

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Step 2.8.2.1

Simplify each term.

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Step 2.8.2.1.1

Multiply $9$ by $- 5$ .

$\frac{63 x^{7} - 45 x^{7} - 5 \cdot 28}{x^{6}}$

Step 2.8.2.1.2

Multiply $- 5$ by $28$ .

$\frac{63 x^{7} - 45 x^{7} - 140}{x^{6}}$

Step 2.8.2.2

Subtract $45 x^{7}$ from $63 x^{7}$ .

$\frac{18 x^{7} - 140}{x^{6}}$

Step 2.8.3

Factor $2$ out of $18 x^{7} - 140$ .

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Step 2.8.3.1

Factor $2$ out of $18 x^{7}$ .

$\frac{2 (9 x^{7}) - 140}{x^{6}}$

Step 2.8.3.2

Factor $2$ out of $- 140$ .

$\frac{2 (9 x^{7}) + 2 (- 70)}{x^{6}}$

Step 2.8.3.3

Factor $2$ out of $2 (9 x^{7}) + 2 (- 70)$ .

$f'' (x) = \frac{2 (9 x^{7} - 70)}{x^{6}}$