Calculus Examples

$\int_{4}^{x^{2}} (3 t^{2} + 2 t) d t$

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

Take the derivative of $\int_{4}^{x^{2}} 3 t^{2} + 2 t d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 3.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.2

Multiply $2$ by $2$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $3$ by $2$ .

$6 x \cdot x^{4} + 2 x (2 x^{2})$

Step 4.2.2

Multiply $x$ by $x^{4}$ by adding the exponents.

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Step 4.2.2.1

Move $x^{4}$ .

$6 (x^{4} x) + 2 x (2 x^{2})$

Step 4.2.2.2

Multiply $x^{4}$ by $x$ .

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Step 4.2.2.2.1

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

$6 (x^{4} x^{1}) + 2 x (2 x^{2})$

Step 4.2.2.2.2

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

$6 x^{4 + 1} + 2 x (2 x^{2})$

Step 4.2.2.3

Add $4$ and $1$ .

$6 x^{5} + 2 x (2 x^{2})$

Step 4.2.3

Multiply $2$ by $2$ .

$6 x^{5} + 4 x \cdot x^{2}$

Step 4.2.4

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

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Step 4.2.4.1

Move $x^{2}$ .

$6 x^{5} + 4 (x^{2} x)$

Step 4.2.4.2

Multiply $x^{2}$ by $x$ .

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Step 4.2.4.2.1

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

$6 x^{5} + 4 (x^{2} x^{1})$

Step 4.2.4.2.2

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

$6 x^{5} + 4 x^{2 + 1}$

Step 4.2.4.3

Add $2$ and $1$ .

$6 x^{5} + 4 x^{3}$