Calculus Examples

$\int_{1}^{14 x^{2}} \frac{1}{t} d t$

Step 1

Take the derivative of $\int_{1}^{14 x^{2}} \frac{1}{t} d t$ with respect to $x$ using Fundamental Theorem of Calculus and the chain rule.

$\frac{d}{d x} [14 x^{2}] \frac{1}{14 x^{2}}$

Step 2

Differentiate.

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Step 2.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x^{2}$ with respect to $x$ is $14 \frac{d}{d x} [x^{2}]$ .

$14 \frac{d}{d x} [x^{2}] \frac{1}{14 x^{2}}$

Step 2.2

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

$14 (2 x) \frac{1}{14 x^{2}}$

Step 2.3

Simplify terms.

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Step 2.3.1

Multiply $2$ by $14$ .

$28 x \frac{1}{14 x^{2}}$

Step 2.3.2

Combine $28$ and $\frac{1}{14 x^{2}}$ .

$x \frac{28}{14 x^{2}}$

Step 2.3.3

Combine $x$ and $\frac{28}{14 x^{2}}$ .

$\frac{x \cdot 28}{14 x^{2}}$

Step 2.3.4

Move $28$ to the left of $x$ .

$\frac{28 \cdot x}{14 x^{2}}$

Step 2.3.5

Cancel the common factor of $28$ and $14$ .

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Step 2.3.5.1

Factor $14$ out of $28 x$ .

$\frac{14 (2 x)}{14 x^{2}}$

Step 2.3.5.2

Cancel the common factors.

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Step 2.3.5.2.1

Factor $14$ out of $14 x^{2}$ .

$\frac{14 (2 x)}{14 (x^{2})}$

Step 2.3.5.2.2

Cancel the common factor.

$\frac{14 (2 x)}{14 x^{2}}$

Step 2.3.5.2.3

Rewrite the expression.

$\frac{2 x}{x^{2}}$

Step 2.3.6

Cancel the common factor of $x$ and $x^{2}$ .

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Step 2.3.6.1

Factor $x$ out of $2 x$ .

$\frac{x \cdot 2}{x^{2}}$

Step 2.3.6.2

Cancel the common factors.

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Step 2.3.6.2.1

Factor $x$ out of $x^{2}$ .

$\frac{x \cdot 2}{x \cdot x}$

Step 2.3.6.2.2

Cancel the common factor.

$\frac{x \cdot 2}{x \cdot x}$

Step 2.3.6.2.3

Rewrite the expression.

$\frac{2}{x}$