Calculus Examples

$\frac{7 e^{7 x}}{x^{2}} - \frac{2 e^{7 x}}{x^{3}}$

Step 1

To write $\frac{7 e^{7 x}}{x^{2}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{7 e^{7 x}}{x^{2}} \cdot \frac{x}{x} - \frac{2 e^{7 x}}{x^{3}}$

Step 2

Write each expression with a common denominator of $x^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1

Multiply $\frac{7 e^{7 x}}{x^{2}}$ by $\frac{x}{x}$ .

$\frac{7 e^{7 x} x}{x^{2} x} - \frac{2 e^{7 x}}{x^{3}}$

Step 2.2

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

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Step 2.2.1

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

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Step 2.2.1.1

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

$\frac{7 e^{7 x} x}{x^{2} x^{1}} - \frac{2 e^{7 x}}{x^{3}}$

Step 2.2.1.2

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

$\frac{7 e^{7 x} x}{x^{2 + 1}} - \frac{2 e^{7 x}}{x^{3}}$

Step 2.2.2

Add $2$ and $1$ .

$\frac{7 e^{7 x} x}{x^{3}} - \frac{2 e^{7 x}}{x^{3}}$

Step 3

Combine the numerators over the common denominator.

$\frac{7 e^{7 x} x - 2 e^{7 x}}{x^{3}}$

Step 4

Factor $e^{7 x}$ out of $7 e^{7 x} x - 2 e^{7 x}$ .

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Step 4.1

Factor $e^{7 x}$ out of $7 e^{7 x} x$ .

$\frac{e^{7 x} (7 x) - 2 e^{7 x}}{x^{3}}$

Step 4.2

Factor $e^{7 x}$ out of $- 2 e^{7 x}$ .

$\frac{e^{7 x} (7 x) + e^{7 x} \cdot - 2}{x^{3}}$

Step 4.3

Factor $e^{7 x}$ out of $e^{7 x} (7 x) + e^{7 x} \cdot - 2$ .

$\frac{e^{7 x} (7 x - 2)}{x^{3}}$