Calculus Examples

$\frac{6 x^{2}}{{(2 x^{3} + 7)}^{\frac{3}{2}}}$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 x^{2}}{{(2 x^{3} + 7)}^{\frac{3}{2}}}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{x^{2}}{{(2 x^{3} + 7)}^{\frac{3}{2}}}]$ .

$6 \frac{d}{d x} [\frac{x^{2}}{{(2 x^{3} + 7)}^{\frac{3}{2}}}]$

Step 2

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 $f (x) = x^{2}$ and $g (x) = {(2 x^{3} + 7)}^{\frac{3}{2}}$ .

$6 \frac{{(2 x^{3} + 7)}^{\frac{3}{2}} \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [{(2 x^{3} + 7)}^{\frac{3}{2}}]}{{({(2 x^{3} + 7)}^{\frac{3}{2}})}^{2}}$

Step 3

Differentiate using the Power Rule.

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Step 3.1

Multiply the exponents in ${({(2 x^{3} + 7)}^{\frac{3}{2}})}^{2}$ .

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Step 3.1.1

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

$6 \frac{{(2 x^{3} + 7)}^{\frac{3}{2}} \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [{(2 x^{3} + 7)}^{\frac{3}{2}}]}{{(2 x^{3} + 7)}^{\frac{3}{2} \cdot 2}}$

Step 3.1.2

Cancel the common factor of $2$ .

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Step 3.1.2.1

Cancel the common factor.

$6 \frac{{(2 x^{3} + 7)}^{\frac{3}{2}} \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [{(2 x^{3} + 7)}^{\frac{3}{2}}]}{{(2 x^{3} + 7)}^{\frac{3}{2} \cdot 2}}$

Step 3.1.2.2

Rewrite the expression.

$6 \frac{{(2 x^{3} + 7)}^{\frac{3}{2}} \frac{d}{d x} [x^{2}] - x^{2} \frac{d}{d x} [{(2 x^{3} + 7)}^{\frac{3}{2}}]}{{(2 x^{3} + 7)}^{3}}$

Step 3.2

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

$6 \frac{{(2 x^{3} + 7)}^{\frac{3}{2}} (2 x) - x^{2} \frac{d}{d x} [{(2 x^{3} + 7)}^{\frac{3}{2}}]}{{(2 x^{3} + 7)}^{3}}$

Step 3.3

Move $2$ to the left of ${(2 x^{3} + 7)}^{\frac{3}{2}}$ .

$6 \frac{2 \cdot {(2 x^{3} + 7)}^{\frac{3}{2}} x - x^{2} \frac{d}{d x} [{(2 x^{3} + 7)}^{\frac{3}{2}}]}{{(2 x^{3} + 7)}^{3}}$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{3}{2}}$ and $g (x) = 2 x^{3} + 7$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $2 x^{3} + 7$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - x^{2} (\frac{d}{d u} [u^{\frac{3}{2}}] \frac{d}{d x} [2 x^{3} + 7])}{{(2 x^{3} + 7)}^{3}}$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{3}{2}$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - x^{2} (\frac{3}{2} u^{\frac{3}{2} - 1} \frac{d}{d x} [2 x^{3} + 7])}{{(2 x^{3} + 7)}^{3}}$

Step 4.3

Replace all occurrences of $u$ with $2 x^{3} + 7$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - x^{2} (\frac{3}{2} {(2 x^{3} + 7)}^{\frac{3}{2} - 1} \frac{d}{d x} [2 x^{3} + 7])}{{(2 x^{3} + 7)}^{3}}$

Step 5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - x^{2} (\frac{3}{2} {(2 x^{3} + 7)}^{\frac{3}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [2 x^{3} + 7])}{{(2 x^{3} + 7)}^{3}}$

Step 6

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

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - x^{2} (\frac{3}{2} {(2 x^{3} + 7)}^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 7])}{{(2 x^{3} + 7)}^{3}}$

Step 7

Combine the numerators over the common denominator.

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - x^{2} (\frac{3}{2} {(2 x^{3} + 7)}^{\frac{3 - 1 \cdot 2}{2}} \frac{d}{d x} [2 x^{3} + 7])}{{(2 x^{3} + 7)}^{3}}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $- 1$ by $2$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - x^{2} (\frac{3}{2} {(2 x^{3} + 7)}^{\frac{3 - 2}{2}} \frac{d}{d x} [2 x^{3} + 7])}{{(2 x^{3} + 7)}^{3}}$

Step 8.2

Subtract $2$ from $3$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - x^{2} (\frac{3}{2} {(2 x^{3} + 7)}^{\frac{1}{2}} \frac{d}{d x} [2 x^{3} + 7])}{{(2 x^{3} + 7)}^{3}}$

Step 9

Combine fractions.

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Step 9.1

Combine $\frac{3}{2}$ and ${(2 x^{3} + 7)}^{\frac{1}{2}}$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - x^{2} (\frac{3 {(2 x^{3} + 7)}^{\frac{1}{2}}}{2} \frac{d}{d x} [2 x^{3} + 7])}{{(2 x^{3} + 7)}^{3}}$

Step 9.2

Combine $\frac{3 {(2 x^{3} + 7)}^{\frac{1}{2}}}{2}$ and $x^{2}$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - \frac{3 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{2} \frac{d}{d x} [2 x^{3} + 7]}{{(2 x^{3} + 7)}^{3}}$

Step 10

By the Sum Rule, the derivative of $2 x^{3} + 7$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [7]$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - \frac{3 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{2} (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [7])}{{(2 x^{3} + 7)}^{3}}$

Step 11

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

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - \frac{3 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{2} (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [7])}{{(2 x^{3} + 7)}^{3}}$

Step 12

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - \frac{3 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{2} (2 (3 x^{2}) + \frac{d}{d x} [7])}{{(2 x^{3} + 7)}^{3}}$

Step 13

Multiply $3$ by $2$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - \frac{3 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{2} (6 x^{2} + \frac{d}{d x} [7])}{{(2 x^{3} + 7)}^{3}}$

Step 14

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

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - \frac{3 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{2} (6 x^{2} + 0)}{{(2 x^{3} + 7)}^{3}}$

Step 15

Combine fractions.

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Step 15.1

Add $6 x^{2}$ and $0$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - \frac{3 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{2} (6 x^{2})}{{(2 x^{3} + 7)}^{3}}$

Step 15.2

Multiply $6$ by $- 1$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - 6 \frac{3 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{2} x^{2}}{{(2 x^{3} + 7)}^{3}}$

Step 15.3

Combine $- 6$ and $\frac{3 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{2}}{2}$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x + \frac{- 6 (3 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{2})}{2} x^{2}}{{(2 x^{3} + 7)}^{3}}$

Step 15.4

Multiply $3$ by $- 6$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x + \frac{- 18 ({(2 x^{3} + 7)}^{\frac{1}{2}} x^{2})}{2} x^{2}}{{(2 x^{3} + 7)}^{3}}$

Step 15.5

Combine $\frac{- 18 ({(2 x^{3} + 7)}^{\frac{1}{2}} x^{2})}{2}$ and $x^{2}$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x + \frac{- 18 ({(2 x^{3} + 7)}^{\frac{1}{2}} x^{2}) x^{2}}{2}}{{(2 x^{3} + 7)}^{3}}$

Step 16

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

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Step 16.1

Move $x^{2}$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x + \frac{- 18 ({(2 x^{3} + 7)}^{\frac{1}{2}} (x^{2} x^{2}))}{2}}{{(2 x^{3} + 7)}^{3}}$

Step 16.2

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

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x + \frac{- 18 ({(2 x^{3} + 7)}^{\frac{1}{2}} x^{2 + 2})}{2}}{{(2 x^{3} + 7)}^{3}}$

Step 16.3

Add $2$ and $2$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x + \frac{- 18 ({(2 x^{3} + 7)}^{\frac{1}{2}} x^{4})}{2}}{{(2 x^{3} + 7)}^{3}}$

Step 17

Factor $2$ out of $- 18 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4}$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x + \frac{2 (- 9 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4})}{2}}{{(2 x^{3} + 7)}^{3}}$

Step 18

Cancel the common factors.

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Step 18.1

Factor $2$ out of $2$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x + \frac{2 (- 9 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4})}{2 (1)}}{{(2 x^{3} + 7)}^{3}}$

Step 18.2

Cancel the common factor.

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x + \frac{2 (- 9 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4})}{2 \cdot 1}}{{(2 x^{3} + 7)}^{3}}$

Step 18.3

Rewrite the expression.

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x + \frac{- 9 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4}}{1}}{{(2 x^{3} + 7)}^{3}}$

Step 18.4

Divide $- 9 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4}$ by $1$ .

$6 \frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - 9 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4}}{{(2 x^{3} + 7)}^{3}}$

Step 19

Combine $6$ and $\frac{2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - 9 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4}}{{(2 x^{3} + 7)}^{3}}$ .

$\frac{6 (2 {(2 x^{3} + 7)}^{\frac{3}{2}} x - 9 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4})}{{(2 x^{3} + 7)}^{3}}$

Step 20

Simplify.

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Step 20.1

Apply the distributive property.

$\frac{6 (2 {(2 x^{3} + 7)}^{\frac{3}{2}} x) + 6 (- 9 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4})}{{(2 x^{3} + 7)}^{3}}$

Step 20.2

Simplify the numerator.

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Step 20.2.1

Multiply $2$ by $6$ .

$\frac{12 ({(2 x^{3} + 7)}^{\frac{3}{2}} x) + 6 (- 9 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4})}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.2

Multiply $- 9$ by $6$ .

$\frac{12 {(2 x^{3} + 7)}^{\frac{3}{2}} x - 54 ({(2 x^{3} + 7)}^{\frac{1}{2}} x^{4})}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.3

Rewrite $12 {(2 x^{3} + 7)}^{\frac{3}{2}} x - 54 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4}$ in a factored form.

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Step 20.2.3.1

Factor $6 {(2 x^{3} + 7)}^{\frac{1}{2}} x$ out of $12 {(2 x^{3} + 7)}^{\frac{3}{2}} x - 54 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4}$ .

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Step 20.2.3.1.1

Reorder the expression.

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Step 20.2.3.1.1.1

Move ${(2 x^{3} + 7)}^{\frac{3}{2}}$ .

$\frac{12 x {(2 x^{3} + 7)}^{\frac{3}{2}} - 54 {(2 x^{3} + 7)}^{\frac{1}{2}} x^{4}}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.3.1.1.2

Move ${(2 x^{3} + 7)}^{\frac{1}{2}}$ .

$\frac{12 x {(2 x^{3} + 7)}^{\frac{3}{2}} - 54 x^{4} {(2 x^{3} + 7)}^{\frac{1}{2}}}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.3.1.2

Factor $6 {(2 x^{3} + 7)}^{\frac{1}{2}} x$ out of $12 x {(2 x^{3} + 7)}^{\frac{3}{2}}$ .

$\frac{6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (2 {(2 x^{3} + 7)}^{\frac{2}{2}}) - 54 x^{4} {(2 x^{3} + 7)}^{\frac{1}{2}}}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.3.1.3

Factor $6 {(2 x^{3} + 7)}^{\frac{1}{2}} x$ out of $- 54 x^{4} {(2 x^{3} + 7)}^{\frac{1}{2}}$ .

$\frac{6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (2 {(2 x^{3} + 7)}^{\frac{2}{2}}) + 6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (- 9 x^{3})}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.3.1.4

Factor $6 {(2 x^{3} + 7)}^{\frac{1}{2}} x$ out of $6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (2 {(2 x^{3} + 7)}^{\frac{2}{2}}) + 6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (- 9 x^{3})$ .

$\frac{6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (2 {(2 x^{3} + 7)}^{\frac{2}{2}} - 9 x^{3})}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.3.2

Divide $2$ by $2$ .

$\frac{6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (2 {(2 x^{3} + 7)}^{1} - 9 x^{3})}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.3.3

Simplify.

$\frac{6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (2 (2 x^{3} + 7) - 9 x^{3})}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.3.4

Apply the distributive property.

$\frac{6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (2 (2 x^{3}) + 2 \cdot 7 - 9 x^{3})}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.3.5

Multiply $2$ by $2$ .

$\frac{6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (4 x^{3} + 2 \cdot 7 - 9 x^{3})}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.3.6

Multiply $2$ by $7$ .

$\frac{6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (4 x^{3} + 14 - 9 x^{3})}{{(2 x^{3} + 7)}^{3}}$

Step 20.2.3.7

Subtract $9 x^{3}$ from $4 x^{3}$ .

$\frac{6 {(2 x^{3} + 7)}^{\frac{1}{2}} x (- 5 x^{3} + 14)}{{(2 x^{3} + 7)}^{3}}$

Step 20.3

Combine terms.

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Step 20.3.1

Move ${(2 x^{3} + 7)}^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{6 x (- 5 x^{3} + 14)}{{(2 x^{3} + 7)}^{3} {(2 x^{3} + 7)}^{- \frac{1}{2}}}$

Step 20.3.2

Multiply ${(2 x^{3} + 7)}^{3}$ by ${(2 x^{3} + 7)}^{- \frac{1}{2}}$ by adding the exponents.

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Step 20.3.2.1

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

$\frac{6 x (- 5 x^{3} + 14)}{{(2 x^{3} + 7)}^{3 - \frac{1}{2}}}$

Step 20.3.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{6 x (- 5 x^{3} + 14)}{{(2 x^{3} + 7)}^{3 \cdot \frac{2}{2} - \frac{1}{2}}}$

Step 20.3.2.3

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

$\frac{6 x (- 5 x^{3} + 14)}{{(2 x^{3} + 7)}^{\frac{3 \cdot 2}{2} - \frac{1}{2}}}$

Step 20.3.2.4

Combine the numerators over the common denominator.

$\frac{6 x (- 5 x^{3} + 14)}{{(2 x^{3} + 7)}^{\frac{3 \cdot 2 - 1}{2}}}$

Step 20.3.2.5

Simplify the numerator.

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Step 20.3.2.5.1

Multiply $3$ by $2$ .

$\frac{6 x (- 5 x^{3} + 14)}{{(2 x^{3} + 7)}^{\frac{6 - 1}{2}}}$

Step 20.3.2.5.2

Subtract $1$ from $6$ .

$\frac{6 x (- 5 x^{3} + 14)}{{(2 x^{3} + 7)}^{\frac{5}{2}}}$

Step 20.4

Factor $- 1$ out of $- 5 x^{3}$ .

$\frac{6 x (- (5 x^{3}) + 14)}{{(2 x^{3} + 7)}^{\frac{5}{2}}}$

Step 20.5

Rewrite $14$ as $- 1 (- 14)$ .

$\frac{6 x (- (5 x^{3}) - 1 (- 14))}{{(2 x^{3} + 7)}^{\frac{5}{2}}}$

Step 20.6

Factor $- 1$ out of $- (5 x^{3}) - 1 (- 14)$ .

$\frac{6 x (- (5 x^{3} - 14))}{{(2 x^{3} + 7)}^{\frac{5}{2}}}$

Step 20.7

Rewrite $- (5 x^{3} - 14)$ as $- 1 (5 x^{3} - 14)$ .

$\frac{6 x (- 1 (5 x^{3} - 14))}{{(2 x^{3} + 7)}^{\frac{5}{2}}}$

Step 20.8

Move the negative in front of the fraction.

$- \frac{(6 x) (5 x^{3} - 14)}{{(2 x^{3} + 7)}^{\frac{5}{2}}}$

Step 20.9

Reorder factors in $- \frac{(6 x) (5 x^{3} - 14)}{{(2 x^{3} + 7)}^{\frac{5}{2}}}$ .

$- \frac{6 x (5 x^{3} - 14)}{{(2 x^{3} + 7)}^{\frac{5}{2}}}$