Calculus Examples

$\int_{1}^{\sqrt{x}} 18 t^{5} d t$

Step 1

Take the derivative of $\int_{1}^{\sqrt{x}} 18 t^{5} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [\sqrt{x}] (18 {\sqrt{x}}^{5})$

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

$\frac{d}{d x} [x^{\frac{1}{2}}] (18 {\sqrt{x}}^{5})$

Step 2.2

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

$\frac{1}{2} x^{\frac{1}{2} - 1} (18 {\sqrt{x}}^{5})$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (18 {\sqrt{x}}^{5})$

Step 4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} (18 {\sqrt{x}}^{5})$

Step 5

Combine the numerators over the common denominator.

$\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} (18 {\sqrt{x}}^{5})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} x^{\frac{1 - 2}{2}} (18 {\sqrt{x}}^{5})$

Step 6.2

Subtract $2$ from $1$ .

$\frac{1}{2} x^{\frac{- 1}{2}} (18 {\sqrt{x}}^{5})$

Step 7

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}} (18 {\sqrt{x}}^{5})$

Step 8

Simplify.

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Step 8.1

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

$\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}} (18 {\sqrt{x}}^{5})$

Step 8.2

Multiply $\frac{1}{2}$ by $\frac{1}{x^{\frac{1}{2}}}$ .

$\frac{1}{2 x^{\frac{1}{2}}} (18 {\sqrt{x}}^{5})$

Step 9

Rewrite ${\sqrt{x}}^{5}$ as $\sqrt{x^{5}}$ .

$\frac{1}{2 x^{\frac{1}{2}}} (18 \sqrt{x^{5}})$

Step 10

Combine $18$ and $\frac{1}{2 x^{\frac{1}{2}}}$ .

$\frac{18}{2 x^{\frac{1}{2}}} \sqrt{x^{5}}$

Step 11

Combine $\frac{18}{2 x^{\frac{1}{2}}}$ and $\sqrt{x^{5}}$ .

$\frac{18 \sqrt{x^{5}}}{2 x^{\frac{1}{2}}}$

Step 12

Factor $2$ out of $18 \sqrt{x^{5}}$ .

$\frac{2 (9 \sqrt{x^{5}})}{2 x^{\frac{1}{2}}}$

Step 13

Cancel the common factors.

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Step 13.1

Factor $2$ out of $2 x^{\frac{1}{2}}$ .

$\frac{2 (9 \sqrt{x^{5}})}{2 (x^{\frac{1}{2}})}$

Step 13.2

Cancel the common factor.

$\frac{2 (9 \sqrt{x^{5}})}{2 x^{\frac{1}{2}}}$

Step 13.3

Rewrite the expression.

$\frac{9 \sqrt{x^{5}}}{x^{\frac{1}{2}}}$

Step 14

Rewrite $\frac{9 \sqrt{x^{5}}}{x^{\frac{1}{2}}}$ as $\frac{9 (x^{2} \sqrt{x})}{x^{\frac{1}{2}}}$ .

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Step 14.1

Rewrite $x^{5}$ as ${(x^{2})}^{2} x$ .

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Step 14.1.1

Factor out $x^{4}$ .

$\frac{9 \sqrt{x^{4} x}}{x^{\frac{1}{2}}}$

Step 14.1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{9 \sqrt{{(x^{2})}^{2} x}}{x^{\frac{1}{2}}}$

Step 14.2

Pull terms out from under the radical.

$\frac{9 (x^{2} \sqrt{x})}{x^{\frac{1}{2}}}$

Step 15

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

$\frac{9 (x^{2} x^{\frac{1}{2}})}{x^{\frac{1}{2}}}$

Step 16

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

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Step 16.1

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

$\frac{9 x^{2 + \frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 16.2

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

$\frac{9 x^{2 \cdot \frac{2}{2} + \frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 16.3

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

$\frac{9 x^{\frac{2 \cdot 2}{2} + \frac{1}{2}}}{x^{\frac{1}{2}}}$

Step 16.4

Combine the numerators over the common denominator.

$\frac{9 x^{\frac{2 \cdot 2 + 1}{2}}}{x^{\frac{1}{2}}}$

Step 16.5

Simplify the numerator.

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Step 16.5.1

Multiply $2$ by $2$ .

$\frac{9 x^{\frac{4 + 1}{2}}}{x^{\frac{1}{2}}}$

Step 16.5.2

Add $4$ and $1$ .

$\frac{9 x^{\frac{5}{2}}}{x^{\frac{1}{2}}}$

Step 17

Move $x^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$9 x^{\frac{5}{2}} x^{- \frac{1}{2}}$

Step 18

Multiply $x^{\frac{5}{2}}$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 18.1

Move $x^{- \frac{1}{2}}$ .

$9 (x^{- \frac{1}{2}} x^{\frac{5}{2}})$

Step 18.2

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

$9 x^{- \frac{1}{2} + \frac{5}{2}}$

Step 18.3

Combine the numerators over the common denominator.

$9 x^{\frac{- 1 + 5}{2}}$

Step 18.4

Add $- 1$ and $5$ .

$9 x^{\frac{4}{2}}$

Step 18.5

Divide $4$ by $2$ .

$9 x^{2}$