Calculus Examples

$\int_{- 2}^{2 x} (3 t^{2} + 2 t) d t$

Step 1

Take the derivative of $\int_{- 2}^{2 x} 3 t^{2} + 2 t d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [2 x] (3 {(2 x)}^{2} + 2 (2 x))$

Step 2

Differentiate.

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Step 2.1

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

$2 \frac{d}{d x} [x] (3 {(2 x)}^{2} + 2 (2 x))$

Step 2.2

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

$2 \cdot 1 (3 {(2 x)}^{2} + 2 (2 x))$

Step 2.3

Simplify with factoring out.

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Step 2.3.1

Multiply $2$ by $1$ .

$2 (3 {(2 x)}^{2} + 2 (2 x))$

Step 2.3.2

Factor $2$ out of $2 x$ .

$2 (3 {(2 (x))}^{2} + 2 (2 x))$

Step 2.3.3

Simplify the expression.

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Step 2.3.3.1

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

$2 (3 (2^{2} x^{2}) + 2 (2 x))$

Step 2.3.3.2

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

$2 (3 (4 x^{2}) + 2 (2 x))$

Step 2.3.3.3

Multiply $4$ by $3$ .

$2 (12 x^{2} + 2 (2 x))$

Step 2.3.3.4

Multiply $2$ by $2$ .

$2 (12 x^{2} + 4 x)$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$2 (12 x^{2}) + 2 (4 x)$

Step 3.2

Combine terms.

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Step 3.2.1

Multiply $12$ by $2$ .

$24 x^{2} + 2 (4 x)$

Step 3.2.2

Multiply $4$ by $2$ .

$24 x^{2} + 8 x$