Calculus Examples

$\int_{0}^{8} \sqrt{8 x} d x$

Step 1

Rewrite $\frac{d}{d x} [\int_{0}^{8} \sqrt{8 x} d x]$ as $\frac{d}{d x} [\int_{0}^{8} 2 \sqrt{2 x} d x]$ .

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Step 1.1

Rewrite $8 x$ as $2^{2} \cdot (2 x)$ .

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Step 1.1.1

Factor $4$ out of $8$ .

$\frac{d}{d x} [\int_{0}^{8} \sqrt{4 (2) x} d x]$

Step 1.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{d}{d x} [\int_{0}^{8} \sqrt{2^{2} \cdot 2 x} d x]$

Step 1.1.3

Add parentheses.

$\frac{d}{d x} [\int_{0}^{8} \sqrt{2^{2} \cdot (2 x)} d x]$

Step 1.2

Pull terms out from under the radical.

$\frac{d}{d x} [\int_{0}^{8} 2 \sqrt{2 x} d x]$

Step 2

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

$\frac{d}{d x} [\int_{0}^{8} 2 {(2 x)}^{\frac{1}{2}} d x]$

Step 3

Factor $2$ out of $2 x$ .

$\frac{d}{d x} [\int_{0}^{8} 2 {(2 (x))}^{\frac{1}{2}} d x]$

Step 4

Apply the product rule to $2 (x)$ .

$\frac{d}{d x} [\int_{0}^{8} 2 (2^{\frac{1}{2}} x^{\frac{1}{2}}) d x]$

Step 5

Raise $2$ to the power of $1$ .

$\frac{d}{d x} [\int_{0}^{8} 2^{\frac{1}{2}} \cdot 2^{1} x^{\frac{1}{2}} d x]$

Step 6

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

$\frac{d}{d x} [\int_{0}^{8} 2^{\frac{1}{2} + 1} x^{\frac{1}{2}} d x]$

Step 7

Simplify the expression.

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Step 7.1

Write $1$ as a fraction with a common denominator.

$\frac{d}{d x} [\int_{0}^{8} 2^{\frac{1}{2} + \frac{2}{2}} x^{\frac{1}{2}} d x]$

Step 7.2

Combine the numerators over the common denominator.

$\frac{d}{d x} [\int_{0}^{8} 2^{\frac{1 + 2}{2}} x^{\frac{1}{2}} d x]$

Step 7.3

Add $1$ and $2$ .

$\frac{d}{d x} [\int_{0}^{8} 2^{\frac{3}{2}} x^{\frac{1}{2}} d x]$

Step 8

Once $\int_{0}^{8} 2^{\frac{3}{2}} x^{\frac{1}{2}} d x$ has been evaluated, it will be constant with respect to $x$ , so the derivative of $\int_{0}^{8} 2^{\frac{3}{2}} x^{\frac{1}{2}} d x$ with respect to $x$ is $0$ .

$0$