Calculus Examples

$y = 5 x^{\frac{2}{5}} - \frac{3}{x^{4}} + 6 \sqrt[3]{x^{2}}$

Step 1

By the Sum Rule, the derivative of $5 x^{\frac{2}{5}} - \frac{3}{x^{4}} + 6 \sqrt[3]{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [5 x^{\frac{2}{5}}] + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$ .

$\frac{d}{d x} [5 x^{\frac{2}{5}}] + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2

Evaluate $\frac{d}{d x} [5 x^{\frac{2}{5}}]$ .

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{\frac{2}{5}}$ with respect to $x$ is $5 \frac{d}{d x} [x^{\frac{2}{5}}]$ .

$5 \frac{d}{d x} [x^{\frac{2}{5}}] + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{5}$ .

$5 (\frac{2}{5} x^{\frac{2}{5} - 1}) + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$5 (\frac{2}{5} x^{\frac{2}{5} - 1 \cdot \frac{5}{5}}) + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.4

Combine $- 1$ and $\frac{5}{5}$ .

$5 (\frac{2}{5} x^{\frac{2}{5} + \frac{- 1 \cdot 5}{5}}) + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.5

Combine the numerators over the common denominator.

$5 (\frac{2}{5} x^{\frac{2 - 1 \cdot 5}{5}}) + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $5$ .

$5 (\frac{2}{5} x^{\frac{2 - 5}{5}}) + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.6.2

Subtract $5$ from $2$ .

$5 (\frac{2}{5} x^{\frac{- 3}{5}}) + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.7

Move the negative in front of the fraction.

$5 (\frac{2}{5} x^{- \frac{3}{5}}) + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.8

Combine $\frac{2}{5}$ and $x^{- \frac{3}{5}}$ .

$5 \frac{2 x^{- \frac{3}{5}}}{5} + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.9

Combine $5$ and $\frac{2 x^{- \frac{3}{5}}}{5}$ .

$\frac{5 (2 x^{- \frac{3}{5}})}{5} + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.10

Multiply $2$ by $5$ .

$\frac{10 x^{- \frac{3}{5}}}{5} + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.11

Move $x^{- \frac{3}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{10}{5 x^{\frac{3}{5}}} + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.12

Factor $5$ out of $10$ .

$\frac{5 \cdot 2}{5 x^{\frac{3}{5}}} + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.13

Cancel the common factors.

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Step 2.13.1

Factor $5$ out of $5 x^{\frac{3}{5}}$ .

$\frac{5 \cdot 2}{5 (x^{\frac{3}{5}})} + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.13.2

Cancel the common factor.

$\frac{5 \cdot 2}{5 x^{\frac{3}{5}}} + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 2.13.3

Rewrite the expression.

$\frac{2}{x^{\frac{3}{5}}} + \frac{d}{d x} [- \frac{3}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3

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

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- \frac{3}{x^{4}}$ with respect to $x$ is $- 3 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$\frac{2}{x^{\frac{3}{5}}} - 3 \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3.2

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

$\frac{2}{x^{\frac{3}{5}}} - 3 \frac{d}{d x} [{(x^{4})}^{- 1}] + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

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 $g (x) = x^{4}$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $x^{4}$ .

$\frac{2}{x^{\frac{3}{5}}} - 3 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3.3.2

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

$\frac{2}{x^{\frac{3}{5}}} - 3 (- u^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3.3.3

Replace all occurrences of $u$ with $x^{4}$ .

$\frac{2}{x^{\frac{3}{5}}} - 3 (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3.4

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

$\frac{2}{x^{\frac{3}{5}}} - 3 (- {(x^{4})}^{- 2} (4 x^{3})) + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3.5

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

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Step 3.5.1

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

$\frac{2}{x^{\frac{3}{5}}} - 3 (- x^{4 \cdot - 2} (4 x^{3})) + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3.5.2

Multiply $4$ by $- 2$ .

$\frac{2}{x^{\frac{3}{5}}} - 3 (- x^{- 8} (4 x^{3})) + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3.6

Multiply $4$ by $- 1$ .

$\frac{2}{x^{\frac{3}{5}}} - 3 (- 4 x^{- 8} x^{3}) + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3.7

Multiply $x^{- 8}$ by $x^{3}$ by adding the exponents.

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Step 3.7.1

Move $x^{3}$ .

$\frac{2}{x^{\frac{3}{5}}} - 3 (- 4 (x^{3} x^{- 8})) + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3.7.2

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

$\frac{2}{x^{\frac{3}{5}}} - 3 (- 4 x^{3 - 8}) + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3.7.3

Subtract $8$ from $3$ .

$\frac{2}{x^{\frac{3}{5}}} - 3 (- 4 x^{- 5}) + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 3.8

Multiply $- 4$ by $- 3$ .

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + \frac{d}{d x} [6 \sqrt[3]{x^{2}}]$

Step 4

Evaluate $\frac{d}{d x} [6 \sqrt[3]{x^{2}}]$ .

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + \frac{d}{d x} [6 x^{\frac{2}{3}}]$

Step 4.2

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

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + 6 \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 4.3

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

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + 6 (\frac{2}{3} x^{\frac{2}{3} - 1})$

Step 4.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + 6 (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 4.5

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

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + 6 (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})$

Step 4.6

Combine the numerators over the common denominator.

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + 6 (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Multiply $- 1$ by $3$ .

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + 6 (\frac{2}{3} x^{\frac{2 - 3}{3}})$

Step 4.7.2

Subtract $3$ from $2$ .

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + 6 (\frac{2}{3} x^{\frac{- 1}{3}})$

Step 4.8

Move the negative in front of the fraction.

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + 6 (\frac{2}{3} x^{- \frac{1}{3}})$

Step 4.9

Combine $\frac{2}{3}$ and $x^{- \frac{1}{3}}$ .

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + 6 \frac{2 x^{- \frac{1}{3}}}{3}$

Step 4.10

Combine $6$ and $\frac{2 x^{- \frac{1}{3}}}{3}$ .

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + \frac{6 (2 x^{- \frac{1}{3}})}{3}$

Step 4.11

Multiply $2$ by $6$ .

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + \frac{12 x^{- \frac{1}{3}}}{3}$

Step 4.12

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

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + \frac{12}{3 x^{\frac{1}{3}}}$

Step 4.13

Factor $3$ out of $12$ .

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + \frac{3 \cdot 4}{3 x^{\frac{1}{3}}}$

Step 4.14

Cancel the common factors.

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Step 4.14.1

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + \frac{3 \cdot 4}{3 (x^{\frac{1}{3}})}$

Step 4.14.2

Cancel the common factor.

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + \frac{3 \cdot 4}{3 x^{\frac{1}{3}}}$

Step 4.14.3

Rewrite the expression.

$\frac{2}{x^{\frac{3}{5}}} + 12 x^{- 5} + \frac{4}{x^{\frac{1}{3}}}$

Step 5

Simplify.

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Step 5.1

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

$\frac{2}{x^{\frac{3}{5}}} + 12 \frac{1}{x^{5}} + \frac{4}{x^{\frac{1}{3}}}$

Step 5.2

Combine $12$ and $\frac{1}{x^{5}}$ .

$\frac{2}{x^{\frac{3}{5}}} + \frac{12}{x^{5}} + \frac{4}{x^{\frac{1}{3}}}$

Step 5.3

Reorder terms.

$\frac{12}{x^{5}} + \frac{2}{x^{\frac{3}{5}}} + \frac{4}{x^{\frac{1}{3}}}$