Calculus Examples

$\int_{0}^{1} 6 x - 2 x^{3} - x \sqrt{x} + 3 e^{x} d x$

Step 1

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

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x \cdot x^{\frac{1}{2}} + 3 e^{x} d x]$

Step 2

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

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Step 2.1

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

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - (x^{\frac{1}{2}} x) + 3 e^{x} d x]$

Step 2.2

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

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Step 2.2.1

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

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - (x^{\frac{1}{2}} x^{1}) + 3 e^{x} d x]$

Step 2.2.2

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

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{1}{2} + 1} + 3 e^{x} d x]$

Step 2.3

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

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{1}{2} + \frac{2}{2}} + 3 e^{x} d x]$

Step 2.4

Combine the numerators over the common denominator.

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{1 + 2}{2}} + 3 e^{x} d x]$

Step 2.5

Add $1$ and $2$ .

$\frac{d}{d x} [\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{3}{2}} + 3 e^{x} d x]$

Step 3

Once $\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{3}{2}} + 3 e^{x} d x$ has been evaluated, it will be constant with respect to $x$ , so the derivative of $\int_{0}^{1} 6 x - 2 x^{3} - x^{\frac{3}{2}} + 3 e^{x} d x$ with respect to $x$ is $0$ .

$0$