Calculus Examples

$F (X) = \frac{e^{X}}{X^{5}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d X} [\frac{f (X)}{g (X)}]$ is $\frac{g (X) \frac{d}{d X} [f (X)] - f (X) \frac{d}{d X} [g (X)]}{{g (X)}^{2}}$ where $f (X) = e^{X}$ and $g (X) = X^{5}$ .

$\frac{X^{5} \frac{d}{d X} [e^{X}] - e^{X} \frac{d}{d X} [X^{5}]}{{(X^{5})}^{2}}$

Step 2

Multiply the exponents in ${(X^{5})}^{2}$ .

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Step 2.1

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

$\frac{X^{5} \frac{d}{d X} [e^{X}] - e^{X} \frac{d}{d X} [X^{5}]}{X^{5 \cdot 2}}$

Step 2.2

Multiply $5$ by $2$ .

$\frac{X^{5} \frac{d}{d X} [e^{X}] - e^{X} \frac{d}{d X} [X^{5}]}{X^{10}}$

Step 3

Differentiate using the Exponential Rule which states that $\frac{d}{d X} [a^{X}]$ is $a^{X} \ln (a)$ where $a$ = $e$ .

$\frac{X^{5} e^{X} - e^{X} \frac{d}{d X} [X^{5}]}{X^{10}}$

Step 4

Differentiate using the Power Rule.

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Step 4.1

Differentiate using the Power Rule which states that $\frac{d}{d X} [X^{n}]$ is $n X^{n - 1}$ where $n = 5$ .

$\frac{X^{5} e^{X} - e^{X} (5 X^{4})}{X^{10}}$

Step 4.2

Multiply $5$ by $- 1$ .

$\frac{X^{5} e^{X} - 5 e^{X} X^{4}}{X^{10}}$

Step 5

Simplify.

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Step 5.1

Reorder factors in $X^{5} e^{X} - 5 e^{X} X^{4}$ .

$\frac{X^{5} e^{X} - 5 X^{4} e^{X}}{X^{10}}$

Step 5.2

Reorder terms.

$\frac{e^{X} X^{5} - 5 e^{X} X^{4}}{X^{10}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Factor $X^{4}$ out of $e^{X} X^{5} - 5 e^{X} X^{4}$ .

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Step 5.3.1.1

Factor $X^{4}$ out of $e^{X} X^{5}$ .

$\frac{X^{4} (e^{X} X) - 5 e^{X} X^{4}}{X^{10}}$

Step 5.3.1.2

Factor $X^{4}$ out of $- 5 e^{X} X^{4}$ .

$\frac{X^{4} (e^{X} X) + X^{4} (- 5 e^{X})}{X^{10}}$

Step 5.3.1.3

Factor $X^{4}$ out of $X^{4} (e^{X} X) + X^{4} (- 5 e^{X})$ .

$\frac{X^{4} (e^{X} X - 5 e^{X})}{X^{10}}$

Step 5.3.2

Factor $e^{X}$ out of $e^{X} X - 5 e^{X}$ .

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Step 5.3.2.1

Factor $e^{X}$ out of $e^{X} X$ .

$\frac{X^{4} (e^{X} (X) - 5 e^{X})}{X^{10}}$

Step 5.3.2.2

Factor $e^{X}$ out of $- 5 e^{X}$ .

$\frac{X^{4} (e^{X} (X) + e^{X} \cdot - 5)}{X^{10}}$

Step 5.3.2.3

Factor $e^{X}$ out of $e^{X} (X) + e^{X} \cdot - 5$ .

$\frac{X^{4} (e^{X} (X - 5))}{X^{10}}$

$\frac{X^{4} e^{X} (X - 5)}{X^{10}}$

Step 5.4

Cancel the common factor of $X^{4}$ and $X^{10}$ .

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Step 5.4.1

Factor $X^{4}$ out of $X^{4} e^{X} (X - 5)$ .

$\frac{X^{4} (e^{X} (X - 5))}{X^{10}}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $X^{4}$ out of $X^{10}$ .

$\frac{X^{4} (e^{X} (X - 5))}{X^{4} X^{6}}$

Step 5.4.2.2

Cancel the common factor.

$\frac{X^{4} (e^{X} (X - 5))}{X^{4} X^{6}}$

Step 5.4.2.3

Rewrite the expression.

$\frac{e^{X} (X - 5)}{X^{6}}$