Calculus Examples

$y = (\sqrt[5]{x} - 2 \sqrt[4]{x} + 1) (x^{3} - 5 x - 7)$

Step 1

Apply basic rules of exponents.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$\frac{d}{d x} [(x^{\frac{1}{5}} - 2 \sqrt[4]{x} + 1) (x^{3} - 5 x - 7)]$

Step 1.2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x}$ as $x^{\frac{1}{4}}$ .

$\frac{d}{d x} [(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (x^{3} - 5 x - 7)]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1$ and $g (x) = x^{3} - 5 x - 7$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) \frac{d}{d x} [x^{3} - 5 x - 7] + (x^{3} - 5 x - 7) \frac{d}{d x} [x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{3} - 5 x - 7$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 7]$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 7]) + (x^{3} - 5 x - 7) \frac{d}{d x} [x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1]$

Step 3.2

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

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- 7]) + (x^{3} - 5 x - 7) \frac{d}{d x} [x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1]$

Step 3.3

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

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]) + (x^{3} - 5 x - 7) \frac{d}{d x} [x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1]$

Step 3.4

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

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5 \cdot 1 + \frac{d}{d x} [- 7]) + (x^{3} - 5 x - 7) \frac{d}{d x} [x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1]$

Step 3.5

Multiply $- 5$ by $1$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5 + \frac{d}{d x} [- 7]) + (x^{3} - 5 x - 7) \frac{d}{d x} [x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1]$

Step 3.6

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

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5 + 0) + (x^{3} - 5 x - 7) \frac{d}{d x} [x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1]$

Step 3.7

Add $3 x^{2} - 5$ and $0$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) \frac{d}{d x} [x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1]$

Step 3.8

By the Sum Rule, the derivative of $x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{1}{5}}] + \frac{d}{d x} [- 2 x^{\frac{1}{4}}] + \frac{d}{d x} [1]$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{d}{d x} [x^{\frac{1}{5}}] + \frac{d}{d x} [- 2 x^{\frac{1}{4}}] + \frac{d}{d x} [1])$

Step 3.9

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

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5} x^{\frac{1}{5} - 1} + \frac{d}{d x} [- 2 x^{\frac{1}{4}}] + \frac{d}{d x} [1])$

Step 4

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

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5} x^{\frac{1}{5} - 1 \cdot \frac{5}{5}} + \frac{d}{d x} [- 2 x^{\frac{1}{4}}] + \frac{d}{d x} [1])$

Step 5

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

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5} x^{\frac{1}{5} + \frac{- 1 \cdot 5}{5}} + \frac{d}{d x} [- 2 x^{\frac{1}{4}}] + \frac{d}{d x} [1])$

Step 6

Combine the numerators over the common denominator.

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5} x^{\frac{1 - 1 \cdot 5}{5}} + \frac{d}{d x} [- 2 x^{\frac{1}{4}}] + \frac{d}{d x} [1])$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $5$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5} x^{\frac{1 - 5}{5}} + \frac{d}{d x} [- 2 x^{\frac{1}{4}}] + \frac{d}{d x} [1])$

Step 7.2

Subtract $5$ from $1$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5} x^{\frac{- 4}{5}} + \frac{d}{d x} [- 2 x^{\frac{1}{4}}] + \frac{d}{d x} [1])$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5} x^{- \frac{4}{5}} + \frac{d}{d x} [- 2 x^{\frac{1}{4}}] + \frac{d}{d x} [1])$

Step 8.2

Combine $\frac{1}{5}$ and $x^{- \frac{4}{5}}$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{x^{- \frac{4}{5}}}{5} + \frac{d}{d x} [- 2 x^{\frac{1}{4}}] + \frac{d}{d x} [1])$

Step 8.3

Move $x^{- \frac{4}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} + \frac{d}{d x} [- 2 x^{\frac{1}{4}}] + \frac{d}{d x} [1])$

Step 9

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{\frac{1}{4}}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{\frac{1}{4}}]$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - 2 \frac{d}{d x} [x^{\frac{1}{4}}] + \frac{d}{d x} [1])$

Step 10

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

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - 2 (\frac{1}{4} x^{\frac{1}{4} - 1}) + \frac{d}{d x} [1])$

Step 11

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - 2 (\frac{1}{4} x^{\frac{1}{4} - 1 \cdot \frac{4}{4}}) + \frac{d}{d x} [1])$

Step 12

Combine $- 1$ and $\frac{4}{4}$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - 2 (\frac{1}{4} x^{\frac{1}{4} + \frac{- 1 \cdot 4}{4}}) + \frac{d}{d x} [1])$

Step 13

Combine the numerators over the common denominator.

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - 2 (\frac{1}{4} x^{\frac{1 - 1 \cdot 4}{4}}) + \frac{d}{d x} [1])$

Step 14

Simplify the numerator.

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Step 14.1

Multiply $- 1$ by $4$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - 2 (\frac{1}{4} x^{\frac{1 - 4}{4}}) + \frac{d}{d x} [1])$

Step 14.2

Subtract $4$ from $1$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - 2 (\frac{1}{4} x^{\frac{- 3}{4}}) + \frac{d}{d x} [1])$

Step 15

Move the negative in front of the fraction.

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - 2 (\frac{1}{4} x^{- \frac{3}{4}}) + \frac{d}{d x} [1])$

Step 16

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

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - 2 \frac{x^{- \frac{3}{4}}}{4} + \frac{d}{d x} [1])$

Step 17

Combine $- 2$ and $\frac{x^{- \frac{3}{4}}}{4}$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} + \frac{- 2 x^{- \frac{3}{4}}}{4} + \frac{d}{d x} [1])$

Step 18

Move $x^{- \frac{3}{4}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} + \frac{- 2}{4 x^{\frac{3}{4}}} + \frac{d}{d x} [1])$

Step 19

Factor $2$ out of $- 2$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} + \frac{2 (- 1)}{4 x^{\frac{3}{4}}} + \frac{d}{d x} [1])$

Step 20

Cancel the common factors.

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Step 20.1

Factor $2$ out of $4 x^{\frac{3}{4}}$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} + \frac{2 (- 1)}{2 (2 x^{\frac{3}{4}})} + \frac{d}{d x} [1])$

Step 20.2

Cancel the common factor.

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} + \frac{2 \cdot - 1}{2 (2 x^{\frac{3}{4}})} + \frac{d}{d x} [1])$

Step 20.3

Rewrite the expression.

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} + \frac{- 1}{2 x^{\frac{3}{4}}} + \frac{d}{d x} [1])$

Step 21

Move the negative in front of the fraction.

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - \frac{1}{2 x^{\frac{3}{4}}} + \frac{d}{d x} [1])$

Step 22

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - \frac{1}{2 x^{\frac{3}{4}}} + 0)$

Step 23

Simplify the expression.

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Step 23.1

Add $\frac{1}{5 x^{\frac{4}{5}}} - \frac{1}{2 x^{\frac{3}{4}}}$ and $0$ .

$(x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) (3 x^{2} - 5) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - \frac{1}{2 x^{\frac{3}{4}}})$

Step 23.2

Reorder terms.

$(3 x^{2} - 5) (x^{\frac{1}{5}} - 2 x^{\frac{1}{4}} + 1) + (x^{3} - 5 x - 7) (\frac{1}{5 x^{\frac{4}{5}}} - \frac{1}{2 x^{\frac{3}{4}}})$