Calculus Examples

$\frac{1 - 6 x}{e^{3 x}}$

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 $f (x) = 1 - 6 x$ and $g (x) = e^{3 x}$ .

$\frac{e^{3 x} \frac{d}{d x} [1 - 6 x] - (1 - 6 x) \frac{d}{d x} [e^{3 x}]}{{(e^{3 x})}^{2}}$

Step 2

Differentiate.

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Step 2.1

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

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Step 2.1.1

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

$\frac{e^{3 x} \frac{d}{d x} [1 - 6 x] - (1 - 6 x) \frac{d}{d x} [e^{3 x}]}{e^{3 x \cdot 2}}$

Step 2.1.2

Multiply $2$ by $3$ .

$\frac{e^{3 x} \frac{d}{d x} [1 - 6 x] - (1 - 6 x) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 2.2

By the Sum Rule, the derivative of $1 - 6 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 6 x]$ .

$\frac{e^{3 x} (\frac{d}{d x} [1] + \frac{d}{d x} [- 6 x]) - (1 - 6 x) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 2.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{e^{3 x} (0 + \frac{d}{d x} [- 6 x]) - (1 - 6 x) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 2.4

Add $0$ and $\frac{d}{d x} [- 6 x]$ .

$\frac{e^{3 x} \frac{d}{d x} [- 6 x] - (1 - 6 x) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 2.5

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$\frac{e^{3 x} (- 6 \frac{d}{d x} [x]) - (1 - 6 x) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 2.6

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

$\frac{e^{3 x} (- 6 \cdot 1) - (1 - 6 x) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 2.7

Simplify the expression.

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Step 2.7.1

Multiply $- 6$ by $1$ .

$\frac{e^{3 x} \cdot - 6 - (1 - 6 x) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

Step 2.7.2

Move $- 6$ to the left of $e^{3 x}$ .

$\frac{- 6 \cdot e^{3 x} - (1 - 6 x) \frac{d}{d x} [e^{3 x}]}{e^{6 x}}$

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 $g (x) = 3 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $3 x$ .

$\frac{- 6 e^{3 x} - (1 - 6 x) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [3 x])}{e^{6 x}}$

Step 3.2

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

$\frac{- 6 e^{3 x} - (1 - 6 x) (e^{u} \frac{d}{d x} [3 x])}{e^{6 x}}$

Step 3.3

Replace all occurrences of $u$ with $3 x$ .

$\frac{- 6 e^{3 x} - (1 - 6 x) (e^{3 x} \frac{d}{d x} [3 x])}{e^{6 x}}$

Step 4

Differentiate.

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Step 4.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$\frac{- 6 e^{3 x} - (1 - 6 x) (e^{3 x} (3 \frac{d}{d x} [x]))}{e^{6 x}}$

Step 4.2

Multiply $3$ by $- 1$ .

$\frac{- 6 e^{3 x} - 3 (1 - 6 x) (e^{3 x} (\frac{d}{d x} [x]))}{e^{6 x}}$

Step 4.3

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

$\frac{- 6 e^{3 x} - 3 (1 - 6 x) (e^{3 x} \cdot 1)}{e^{6 x}}$

Step 4.4

Multiply $e^{3 x}$ by $1$ .

$\frac{- 6 e^{3 x} - 3 (1 - 6 x) e^{3 x}}{e^{6 x}}$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{- 6 e^{3 x} + (- 3 \cdot 1 - 3 (- 6 x)) e^{3 x}}{e^{6 x}}$

Step 5.2

Apply the distributive property.

$\frac{- 6 e^{3 x} - 3 \cdot 1 e^{3 x} - 3 (- 6 x) e^{3 x}}{e^{6 x}}$

Step 5.3

Simplify the numerator.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply $- 3$ by $1$ .

$\frac{- 6 e^{3 x} - 3 e^{3 x} - 3 (- 6 x) e^{3 x}}{e^{6 x}}$

Step 5.3.1.2

Multiply $- 6$ by $- 3$ .

$\frac{- 6 e^{3 x} - 3 e^{3 x} + 18 x e^{3 x}}{e^{6 x}}$

Step 5.3.2

Subtract $3 e^{3 x}$ from $- 6 e^{3 x}$ .

$\frac{- 9 e^{3 x} + 18 x e^{3 x}}{e^{6 x}}$

Step 5.4

Reorder terms.

$\frac{18 e^{3 x} x - 9 e^{3 x}}{e^{6 x}}$

Step 5.5

Factor $9 e^{3 x}$ out of $18 e^{3 x} x - 9 e^{3 x}$ .

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Step 5.5.1

Factor $9 e^{3 x}$ out of $18 e^{3 x} x$ .

$\frac{9 e^{3 x} (2 x) - 9 e^{3 x}}{e^{6 x}}$

Step 5.5.2

Factor $9 e^{3 x}$ out of $- 9 e^{3 x}$ .

$\frac{9 e^{3 x} (2 x) + 9 e^{3 x} (- 1)}{e^{6 x}}$

Step 5.5.3

Factor $9 e^{3 x}$ out of $9 e^{3 x} (2 x) + 9 e^{3 x} (- 1)$ .

$\frac{9 e^{3 x} (2 x - 1)}{e^{6 x}}$

Step 5.6

Cancel the common factor of $e^{3 x}$ and $e^{6 x}$ .

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Step 5.6.1

Factor $e^{6 x}$ out of $9 e^{3 x} (2 x - 1)$ .

$\frac{e^{6 x} (9 e^{- 3 x} (2 x - 1))}{e^{6 x}}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Multiply by $1$ .

$\frac{e^{6 x} (9 e^{- 3 x} (2 x - 1))}{e^{6 x} \cdot 1}$

Step 5.6.2.2

Cancel the common factor.

$\frac{e^{6 x} (9 e^{- 3 x} (2 x - 1))}{e^{6 x} \cdot 1}$

Step 5.6.2.3

Rewrite the expression.

$\frac{9 e^{- 3 x} (2 x - 1)}{1}$

Step 5.6.2.4

Divide $9 e^{- 3 x} (2 x - 1)$ by $1$ .

$9 e^{- 3 x} (2 x - 1)$

Step 5.7

Apply the distributive property.

$9 e^{- 3 x} (2 x) + 9 e^{- 3 x} \cdot - 1$

Step 5.8

Rewrite using the commutative property of multiplication.

$9 \cdot 2 e^{- 3 x} x + 9 e^{- 3 x} \cdot - 1$

Step 5.9

Multiply $- 1$ by $9$ .

$9 \cdot 2 e^{- 3 x} x - 9 e^{- 3 x}$

Step 5.10

Multiply $9$ by $2$ .

$18 e^{- 3 x} x - 9 e^{- 3 x}$

Step 5.11

Reorder factors in $18 e^{- 3 x} x - 9 e^{- 3 x}$ .

$18 x e^{- 3 x} - 9 e^{- 3 x}$