Calculus Examples

$f (x) = 5^{3} \sqrt{x^{4}} + \frac{2}{x^{2}} - \frac{1}{3 x^{3}}$

Step 1

Differentiate using the Sum Rule.

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Step 1.1

Raise $5$ to the power of $3$ .

$\frac{d}{d x} [125 \sqrt{x^{4}} + \frac{2}{x^{2}} - \frac{1}{3 x^{3}}]$

Step 1.2

By the Sum Rule, the derivative of $125 \sqrt{x^{4}} + \frac{2}{x^{2}} - \frac{1}{3 x^{3}}$ with respect to $x$ is $\frac{d}{d x} [125 \sqrt{x^{4}}] + \frac{d}{d x} [\frac{2}{x^{2}}] + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$ .

$\frac{d}{d x} [125 \sqrt{x^{4}}] + \frac{d}{d x} [\frac{2}{x^{2}}] + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 2

Evaluate $\frac{d}{d x} [125 \sqrt{x^{4}}]$ .

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4}}$ as $x^{\frac{4}{2}}$ .

$\frac{d}{d x} [125 x^{\frac{4}{2}}] + \frac{d}{d x} [\frac{2}{x^{2}}] + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 2.2

Divide $4$ by $2$ .

$\frac{d}{d x} [125 x^{2}] + \frac{d}{d x} [\frac{2}{x^{2}}] + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 2.3

Since $125$ is constant with respect to $x$ , the derivative of $125 x^{2}$ with respect to $x$ is $125 \frac{d}{d x} [x^{2}]$ .

$125 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{2}{x^{2}}] + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 2.4

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

$125 (2 x) + \frac{d}{d x} [\frac{2}{x^{2}}] + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 2.5

Multiply $2$ by $125$ .

$250 x + \frac{d}{d x} [\frac{2}{x^{2}}] + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3

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

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Step 3.1

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

$250 x + 2 \frac{d}{d x} [\frac{1}{x^{2}}] + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3.2

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

$250 x + 2 \frac{d}{d x} [{(x^{2})}^{- 1}] + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

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 $g (x) = x^{2}$ .

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Step 3.3.1

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

$250 x + 2 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3.3.2

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

$250 x + 2 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3.3.3

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

$250 x + 2 (- {(x^{2})}^{- 2} \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3.4

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

$250 x + 2 (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3.5

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

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Step 3.5.1

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

$250 x + 2 (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3.5.2

Multiply $2$ by $- 2$ .

$250 x + 2 (- x^{- 4} (2 x)) + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3.6

Multiply $2$ by $- 1$ .

$250 x + 2 (- 2 x^{- 4} x) + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3.7

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

$250 x + 2 (- 2 (x^{1} x^{- 4})) + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3.8

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

$250 x + 2 (- 2 x^{1 - 4}) + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3.9

Subtract $4$ from $1$ .

$250 x + 2 (- 2 x^{- 3}) + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 3.10

Multiply $- 2$ by $2$ .

$250 x - 4 x^{- 3} + \frac{d}{d x} [- \frac{1}{3 x^{3}}]$

Step 4

Evaluate $\frac{d}{d x} [- \frac{1}{3 x^{3}}]$ .

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Step 4.1

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

$250 x - 4 x^{- 3} - \frac{1}{3} \frac{d}{d x} [\frac{1}{x^{3}}]$

Step 4.2

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

$250 x - 4 x^{- 3} - \frac{1}{3} \frac{d}{d x} [{(x^{3})}^{- 1}]$

Step 4.3

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 4.3.1

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

$250 x - 4 x^{- 3} - \frac{1}{3} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{3}])$

Step 4.3.2

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

$250 x - 4 x^{- 3} - \frac{1}{3} (- {u_{2}}^{- 2} \frac{d}{d x} [x^{3}])$

Step 4.3.3

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

$250 x - 4 x^{- 3} - \frac{1}{3} (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}])$

Step 4.4

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

$250 x - 4 x^{- 3} - \frac{1}{3} (- {(x^{3})}^{- 2} (3 x^{2}))$

Step 4.5

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

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Step 4.5.1

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

$250 x - 4 x^{- 3} - \frac{1}{3} (- x^{3 \cdot - 2} (3 x^{2}))$

Step 4.5.2

Multiply $3$ by $- 2$ .

$250 x - 4 x^{- 3} - \frac{1}{3} (- x^{- 6} (3 x^{2}))$

Step 4.6

Multiply $3$ by $- 1$ .

$250 x - 4 x^{- 3} - \frac{1}{3} (- 3 x^{- 6} x^{2})$

Step 4.7

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

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Step 4.7.1

Move $x^{2}$ .

$250 x - 4 x^{- 3} - \frac{1}{3} (- 3 (x^{2} x^{- 6}))$

Step 4.7.2

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

$250 x - 4 x^{- 3} - \frac{1}{3} (- 3 x^{2 - 6})$

Step 4.7.3

Subtract $6$ from $2$ .

$250 x - 4 x^{- 3} - \frac{1}{3} (- 3 x^{- 4})$

Step 4.8

Multiply $- 3$ by $- 1$ .

$250 x - 4 x^{- 3} + 3 (\frac{1}{3}) x^{- 4}$

Step 4.9

Combine $3$ and $\frac{1}{3}$ .

$250 x - 4 x^{- 3} + \frac{3}{3} x^{- 4}$

Step 4.10

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

$250 x - 4 x^{- 3} + \frac{3 x^{- 4}}{3}$

Step 4.11

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$250 x - 4 x^{- 3} + \frac{3}{3 x^{4}}$

Step 4.12

Cancel the common factor of $3$ .

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Step 4.12.1

Cancel the common factor.

$250 x - 4 x^{- 3} + \frac{3}{3 x^{4}}$

Step 4.12.2

Rewrite the expression.

$250 x - 4 x^{- 3} + \frac{1}{x^{4}}$

Step 5

Simplify.

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Step 5.1

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

$250 x - 4 \frac{1}{x^{3}} + \frac{1}{x^{4}}$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $- 4$ and $\frac{1}{x^{3}}$ .

$250 x + \frac{- 4}{x^{3}} + \frac{1}{x^{4}}$

Step 5.2.2

Move the negative in front of the fraction.

$250 x - \frac{4}{x^{3}} + \frac{1}{x^{4}}$