Calculus Examples

$\int_{- 2 x}^{2 x} s^{2} d s$

Step 1

Split the integral into two integrals where $c$ is some value between $- 2 x$ and $2 x$ .

$\frac{d}{d x} [\int_{- 2 x}^{c} s^{2} d s + \int_{c}^{2 x} s^{2} d s]$

Step 2

By the Sum Rule, the derivative of $\int_{- 2 x}^{c} s^{2} d s + \int_{c}^{2 x} s^{2} d s$ with respect to $x$ is $\frac{d}{d x} [\int_{- 2 x}^{c} s^{2} d s] + \frac{d}{d x} [\int_{c}^{2 x} s^{2} d s]$ .

$\frac{d}{d x} [\int_{- 2 x}^{c} s^{2} d s] + \frac{d}{d x} [\int_{c}^{2 x} s^{2} d s]$

Step 3

Swap the bounds of integration.

$\frac{d}{d x} [- \int_{c}^{- 2 x} s^{2} d s] + \frac{d}{d x} [\int_{c}^{2 x} s^{2} d s]$

Step 4

Take the derivative of $- \int_{c}^{- 2 x} s^{2} d s$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [- 2 x] (- {(- 2 x)}^{2}) + \frac{d}{d x} [\int_{c}^{2 x} s^{2} d s]$

Step 5

Differentiate.

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Step 5.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$- 2 \frac{d}{d x} [x] (- {(- 2 x)}^{2}) + \frac{d}{d x} [\int_{c}^{2 x} s^{2} d s]$

Step 5.2

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

$- 2 \cdot 1 (- {(- 2 x)}^{2}) + \frac{d}{d x} [\int_{c}^{2 x} s^{2} d s]$

Step 5.3

Multiply $- 2$ by $1$ .

$- 2 (- {(- 2 x)}^{2}) + \frac{d}{d x} [\int_{c}^{2 x} s^{2} d s]$

Step 6

Take the derivative of $\int_{c}^{2 x} s^{2} d s$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$- 2 (- {(- 2 x)}^{2}) + \frac{d}{d x} [2 x] {(2 x)}^{2}$

Step 7

Differentiate.

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Step 7.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$- 2 (- {(- 2 x)}^{2}) + 2 \frac{d}{d x} [x] {(2 x)}^{2}$

Step 7.2

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

$- 2 (- {(- 2 x)}^{2}) + 2 \cdot 1 {(2 x)}^{2}$

Step 7.3

Simplify terms.

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Step 7.3.1

Multiply $2$ by $1$ .

$- 2 (- {(- 2 x)}^{2}) + 2 {(2 x)}^{2}$

Step 7.3.2

Factor $- 2$ out of $- 2 x$ .

$- 2 (- {(- 2 (x))}^{2}) + 2 {(2 x)}^{2}$

Step 7.3.3

Simplify the expression.

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Step 7.3.3.1

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

$- 2 (- ({(- 2)}^{2} x^{2})) + 2 {(2 x)}^{2}$

Step 7.3.3.2

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

$- 2 (- (4 x^{2})) + 2 {(2 x)}^{2}$

Step 7.3.3.3

Multiply $4$ by $- 1$ .

$- 2 (- 4 x^{2}) + 2 {(2 x)}^{2}$

Step 7.3.3.4

Multiply $- 4$ by $- 2$ .

$8 x^{2} + 2 {(2 x)}^{2}$

Step 7.3.4

Factor $2$ out of $2 x$ .

$8 x^{2} + 2 {(2 (x))}^{2}$

Step 7.3.5

Simplify the expression.

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Step 7.3.5.1

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

$8 x^{2} + 2 (2^{2} x^{2})$

Step 7.3.5.2

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

$8 x^{2} + 2 (4 x^{2})$

Step 7.3.5.3

Multiply $4$ by $2$ .

$8 x^{2} + 8 x^{2}$

Step 7.3.6

Add $8 x^{2}$ and $8 x^{2}$ .

$16 x^{2}$