Calculus Examples

$\frac{d}{d x} (\int x^{11} d x)$

Step 1

Cancel the common factor of $d$ .

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Step 1.1

Cancel the common factor.

$\frac{d}{d x} [\frac{d}{d x} \cdot \int x^{11} d x]$

Step 1.2

Rewrite the expression.

$\frac{d}{d x} [\frac{1}{x} \cdot \int x^{11} d x]$

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 $f (x) = \frac{1}{x}$ and $g (x) = \int x^{11} d x$ .

$\frac{1}{x} \frac{d}{d x} [\int x^{11} d x] + \int x^{11} d x \frac{d}{d x} [\frac{1}{x}]$

Step 3

$\int x^{11} d x$ is an antiderivative of $x^{11}$ , so by definition $\frac{d}{d x} [\int x^{11} d x]$ is $x^{11}$ .

$\frac{1}{x} x^{11} + \int x^{11} d x \frac{d}{d x} [\frac{1}{x}]$

Step 4

Simplify terms.

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Step 4.1

Combine $\frac{1}{x}$ and $x^{11}$ .

$\frac{x^{11}}{x} + \int x^{11} d x \frac{d}{d x} [\frac{1}{x}]$

Step 4.2

Cancel the common factor of $x^{11}$ and $x$ .

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Step 4.2.1

Factor $x$ out of $x^{11}$ .

$\frac{x \cdot x^{10}}{x} + \int x^{11} d x \frac{d}{d x} [\frac{1}{x}]$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

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

$\frac{x \cdot x^{10}}{x^{1}} + \int x^{11} d x \frac{d}{d x} [\frac{1}{x}]$

Step 4.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot x^{10}}{x \cdot 1} + \int x^{11} d x \frac{d}{d x} [\frac{1}{x}]$

Step 4.2.2.3

Cancel the common factor.

$\frac{x \cdot x^{10}}{x \cdot 1} + \int x^{11} d x \frac{d}{d x} [\frac{1}{x}]$

Step 4.2.2.4

Rewrite the expression.

$\frac{x^{10}}{1} + \int x^{11} d x \frac{d}{d x} [\frac{1}{x}]$

Step 4.2.2.5

Divide $x^{10}$ by $1$ .

$x^{10} + \int x^{11} d x \frac{d}{d x} [\frac{1}{x}]$

Step 4.3

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$x^{10} + \int x^{11} d x \frac{d}{d x} [x^{- 1}]$

Step 5

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

$x^{10} + \int x^{11} d x (- x^{- 2})$

Step 6

Simplify.

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Step 6.1

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

$x^{10} + \int x^{11} d x (- \frac{1}{x^{2}})$

Step 6.2

Reorder terms.

$x^{10} - \frac{1}{x^{2}} \int x^{11} d x$

Step 6.3

Combine $- \frac{1}{x^{2}}$ and $\int x^{11} d x$ .

$x^{10} - \frac{\int x^{11} d x}{x^{2}}$