Calculus Examples

$f (x) = \frac{9}{x^{2}} + \frac{7}{x^{3}}$

Step 1

By the Sum Rule, the derivative of $\frac{9}{x^{2}} + \frac{7}{x^{3}}$ with respect to $x$ is $\frac{d}{d x} [\frac{9}{x^{2}}] + \frac{d}{d x} [\frac{7}{x^{3}}]$ .

$\frac{d}{d x} [\frac{9}{x^{2}}] + \frac{d}{d x} [\frac{7}{x^{3}}]$

Step 2

Evaluate $\frac{d}{d x} [\frac{9}{x^{2}}]$ .

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Step 2.1

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

$9 \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [\frac{7}{x^{3}}]$

Step 2.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{2}$ .

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Step 2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

$9 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{7}{x^{3}}]$

Step 2.3.2

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

$9 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{7}{x^{3}}]$

Step 2.3.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

$9 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [\frac{7}{x^{3}}]$

Step 2.4

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

$9 (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} [\frac{7}{x^{3}}]$

Step 2.5

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

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Step 2.5.1

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

$9 (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} [\frac{7}{x^{3}}]$

Step 2.5.2

Multiply $2$ by $- 2$ .

$9 (- x^{- 4} (2 x)) + \frac{d}{d x} [\frac{7}{x^{3}}]$

Step 2.6

Multiply $2$ by $- 1$ .

$9 (- 2 x^{- 4} x) + \frac{d}{d x} [\frac{7}{x^{3}}]$

Step 2.7

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

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

Step 2.8

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

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

Step 2.9

Subtract $4$ from $1$ .

$9 (- 2 x^{- 3}) + \frac{d}{d x} [\frac{7}{x^{3}}]$

Step 2.10

Multiply $- 2$ by $9$ .

$- 18 x^{- 3} + \frac{d}{d x} [\frac{7}{x^{3}}]$

Step 3

Evaluate $\frac{d}{d x} [\frac{7}{x^{3}}]$ .

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Step 3.1

Since $7$ is constant with respect to $x$ , the derivative of $\frac{7}{x^{3}}$ with respect to $x$ is $7 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$- 18 x^{- 3} + 7 \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 3.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$- 18 x^{- 3} + 7 \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

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Step 3.3.1

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

$- 18 x^{- 3} + 7 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{3}])$

Step 3.3.2

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

$- 18 x^{- 3} + 7 (- {u_{2}}^{- 2} \frac{d}{d x} [x^{3}])$

Step 3.3.3

Replace all occurrences of $u_{2}$ with $x^{3}$ .

$- 18 x^{- 3} + 7 (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}])$

Step 3.4

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

$- 18 x^{- 3} + 7 (- {(x^{3})}^{- 2} (3 x^{2}))$

Step 3.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 3.5.1

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

$- 18 x^{- 3} + 7 (- x^{3 \cdot - 2} (3 x^{2}))$

Step 3.5.2

Multiply $3$ by $- 2$ .

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

Step 3.6

Multiply $3$ by $- 1$ .

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

Step 3.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 3.7.1

Move $x^{2}$ .

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

Step 3.7.2

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

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

Step 3.7.3

Subtract $6$ from $2$ .

$- 18 x^{- 3} + 7 (- 3 x^{- 4})$

Step 3.8

Multiply $- 3$ by $7$ .

$- 18 x^{- 3} - 21 x^{- 4}$

Step 4

Simplify.

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Step 4.1

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

$- 18 \frac{1}{x^{3}} - 21 x^{- 4}$

Step 4.2

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

$- 18 \frac{1}{x^{3}} - 21 \frac{1}{x^{4}}$

Step 4.3

Combine terms.

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Step 4.3.1

Combine $- 18$ and $\frac{1}{x^{3}}$ .

$\frac{- 18}{x^{3}} - 21 \frac{1}{x^{4}}$

Step 4.3.2

Move the negative in front of the fraction.

$- \frac{18}{x^{3}} - 21 \frac{1}{x^{4}}$

Step 4.3.3

Combine $- 21$ and $\frac{1}{x^{4}}$ .

$- \frac{18}{x^{3}} + \frac{- 21}{x^{4}}$

Step 4.3.4

Move the negative in front of the fraction.

$- \frac{18}{x^{3}} - \frac{21}{x^{4}}$