Calculus Examples

$\int_{- 1}^{x^{2}} e^{t} + t^{3} d t$

Step 1

Take the derivative of $\int_{- 1}^{x^{2}} e^{t} + t^{3} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

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

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

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

Step 2.2

Multiply the exponents in ${(x^{2})}^{3}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $3$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$2 x e^{x^{2}} + 2 x \cdot x^{6}$

Step 3.2

Multiply $x$ by $x^{6}$ by adding the exponents.

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Step 3.2.1

Move $x^{6}$ .

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

Step 3.2.2

Multiply $x^{6}$ by $x$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

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

$2 x e^{x^{2}} + 2 x^{6 + 1}$

Step 3.2.3

Add $6$ and $1$ .

$2 x e^{x^{2}} + 2 x^{7}$

Step 3.3

Reorder terms.

$2 x^{7} + 2 e^{x^{2}} x$

Step 3.4

Reorder factors in $2 x^{7} + 2 e^{x^{2}} x$ .

$2 x^{7} + 2 x e^{x^{2}}$