Calculus Examples

$f (x) = x (x^{8}) (\sqrt{x}) (x^{\frac{7}{8}})$

Step 1

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

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Step 1.1

Multiply $x$ by $x^{8}$ .

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Step 1.1.1

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

$\frac{d}{d x} [x^{1} x^{8} \sqrt{x} x^{\frac{7}{8}}]$

Step 1.1.2

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

$\frac{d}{d x} [x^{1 + 8} \sqrt{x} x^{\frac{7}{8}}]$

Step 1.2

Add $1$ and $8$ .

$\frac{d}{d x} [x^{9} \sqrt{x} x^{\frac{7}{8}}]$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{d}{d x} [x^{9} x^{\frac{1}{2}} x^{\frac{7}{8}}]$

Step 3

Multiply $x^{9}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 3.1

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

$\frac{d}{d x} [x^{9 + \frac{1}{2}} x^{\frac{7}{8}}]$

Step 3.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{d}{d x} [x^{9 \cdot \frac{2}{2} + \frac{1}{2}} x^{\frac{7}{8}}]$

Step 3.3

Combine $9$ and $\frac{2}{2}$ .

$\frac{d}{d x} [x^{\frac{9 \cdot 2}{2} + \frac{1}{2}} x^{\frac{7}{8}}]$

Step 3.4

Combine the numerators over the common denominator.

$\frac{d}{d x} [x^{\frac{9 \cdot 2 + 1}{2}} x^{\frac{7}{8}}]$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $9$ by $2$ .

$\frac{d}{d x} [x^{\frac{18 + 1}{2}} x^{\frac{7}{8}}]$

Step 3.5.2

Add $18$ and $1$ .

$\frac{d}{d x} [x^{\frac{19}{2}} x^{\frac{7}{8}}]$

Step 4

Multiply $x^{\frac{19}{2}}$ by $x^{\frac{7}{8}}$ by adding the exponents.

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Step 4.1

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

$\frac{d}{d x} [x^{\frac{19}{2} + \frac{7}{8}}]$

Step 4.2

To write $\frac{19}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{d}{d x} [x^{\frac{19}{2} \cdot \frac{4}{4} + \frac{7}{8}}]$

Step 4.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.3.1

Multiply $\frac{19}{2}$ by $\frac{4}{4}$ .

$\frac{d}{d x} [x^{\frac{19 \cdot 4}{2 \cdot 4} + \frac{7}{8}}]$

Step 4.3.2

Multiply $2$ by $4$ .

$\frac{d}{d x} [x^{\frac{19 \cdot 4}{8} + \frac{7}{8}}]$

Step 4.4

Combine the numerators over the common denominator.

$\frac{d}{d x} [x^{\frac{19 \cdot 4 + 7}{8}}]$

Step 4.5

Simplify the numerator.

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Step 4.5.1

Multiply $19$ by $4$ .

$\frac{d}{d x} [x^{\frac{76 + 7}{8}}]$

Step 4.5.2

Add $76$ and $7$ .

$\frac{d}{d x} [x^{\frac{83}{8}}]$

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{83}{8}$ .

$\frac{83}{8} x^{\frac{83}{8} - 1}$

Step 6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{83}{8} x^{\frac{83}{8} - 1 \cdot \frac{8}{8}}$

Step 7

Combine $- 1$ and $\frac{8}{8}$ .

$\frac{83}{8} x^{\frac{83}{8} + \frac{- 1 \cdot 8}{8}}$

Step 8

Combine the numerators over the common denominator.

$\frac{83}{8} x^{\frac{83 - 1 \cdot 8}{8}}$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $8$ .

$\frac{83}{8} x^{\frac{83 - 8}{8}}$

Step 9.2

Subtract $8$ from $83$ .

$\frac{83}{8} x^{\frac{75}{8}}$

Step 10

Combine $\frac{83}{8}$ and $x^{\frac{75}{8}}$ .

$\frac{83 x^{\frac{75}{8}}}{8}$