Calculus Examples

$\frac{e^{5 x}}{x^{15}}$

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 $f (x) = e^{5 x}$ and $g (x) = x^{15}$ .

$\frac{x^{15} \frac{d}{d x} [e^{5 x}] - e^{5 x} \frac{d}{d x} [x^{15}]}{{(x^{15})}^{2}}$

Step 2

Multiply the exponents in ${(x^{15})}^{2}$ .

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Step 2.1

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

$\frac{x^{15} \frac{d}{d x} [e^{5 x}] - e^{5 x} \frac{d}{d x} [x^{15}]}{x^{15 \cdot 2}}$

Step 2.2

Multiply $15$ by $2$ .

$\frac{x^{15} \frac{d}{d x} [e^{5 x}] - e^{5 x} \frac{d}{d x} [x^{15}]}{x^{30}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 5 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $5 x$ .

$\frac{x^{15} (\frac{d}{d u} [e^{u}] \frac{d}{d x} [5 x]) - e^{5 x} \frac{d}{d x} [x^{15}]}{x^{30}}$

Step 3.2

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

$\frac{x^{15} (e^{u} \frac{d}{d x} [5 x]) - e^{5 x} \frac{d}{d x} [x^{15}]}{x^{30}}$

Step 3.3

Replace all occurrences of $u$ with $5 x$ .

$\frac{x^{15} (e^{5 x} \frac{d}{d x} [5 x]) - e^{5 x} \frac{d}{d x} [x^{15}]}{x^{30}}$

Step 4

Differentiate.

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Step 4.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$\frac{x^{15} (e^{5 x} (5 \frac{d}{d x} [x])) - e^{5 x} \frac{d}{d x} [x^{15}]}{x^{30}}$

Step 4.2

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

$\frac{x^{15} (e^{5 x} (5 \cdot 1)) - e^{5 x} \frac{d}{d x} [x^{15}]}{x^{30}}$

Step 4.3

Simplify the expression.

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Step 4.3.1

Multiply $5$ by $1$ .

$\frac{x^{15} (e^{5 x} \cdot 5) - e^{5 x} \frac{d}{d x} [x^{15}]}{x^{30}}$

Step 4.3.2

Move $5$ to the left of $e^{5 x}$ .

$\frac{x^{15} (5 \cdot e^{5 x}) - e^{5 x} \frac{d}{d x} [x^{15}]}{x^{30}}$

Step 4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 15$ .

$\frac{x^{15} (5 e^{5 x}) - e^{5 x} (15 x^{14})}{x^{30}}$

Step 4.5

Multiply $15$ by $- 1$ .

$\frac{x^{15} (5 e^{5 x}) - 15 e^{5 x} x^{14}}{x^{30}}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Rewrite using the commutative property of multiplication.

$\frac{5 x^{15} e^{5 x} - 15 e^{5 x} x^{14}}{x^{30}}$

Step 5.1.2

Reorder factors in $5 x^{15} e^{5 x} - 15 e^{5 x} x^{14}$ .

$\frac{5 x^{15} e^{5 x} - 15 x^{14} e^{5 x}}{x^{30}}$

Step 5.2

Reorder terms.

$\frac{5 e^{5 x} x^{15} - 15 e^{5 x} x^{14}}{x^{30}}$

Step 5.3

Factor $5 e^{5 x} x^{14}$ out of $5 e^{5 x} x^{15} - 15 e^{5 x} x^{14}$ .

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Step 5.3.1

Factor $5 e^{5 x} x^{14}$ out of $5 e^{5 x} x^{15}$ .

$\frac{5 e^{5 x} x^{14} (x) - 15 e^{5 x} x^{14}}{x^{30}}$

Step 5.3.2

Factor $5 e^{5 x} x^{14}$ out of $- 15 e^{5 x} x^{14}$ .

$\frac{5 e^{5 x} x^{14} (x) + 5 e^{5 x} x^{14} (- 3)}{x^{30}}$

Step 5.3.3

Factor $5 e^{5 x} x^{14}$ out of $5 e^{5 x} x^{14} (x) + 5 e^{5 x} x^{14} (- 3)$ .

$\frac{5 e^{5 x} x^{14} (x - 3)}{x^{30}}$

Step 5.4

Cancel the common factor of $x^{14}$ and $x^{30}$ .

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Step 5.4.1

Factor $x^{14}$ out of $5 e^{5 x} x^{14} (x - 3)$ .

$\frac{x^{14} (5 e^{5 x} (x - 3))}{x^{30}}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $x^{14}$ out of $x^{30}$ .

$\frac{x^{14} (5 e^{5 x} (x - 3))}{x^{14} x^{16}}$

Step 5.4.2.2

Cancel the common factor.

$\frac{x^{14} (5 e^{5 x} (x - 3))}{x^{14} x^{16}}$

Step 5.4.2.3

Rewrite the expression.

$\frac{5 e^{5 x} (x - 3)}{x^{16}}$