Calculus Examples

$\frac{d}{d x} \cdot \frac{6 x}{e^{x}}$

Step 1

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

$6 \frac{d}{d x} [\frac{x}{e^{x}}]$

Step 2

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

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

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

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Step 3.1.1

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

$6 \frac{e^{x} \frac{d}{d x} [x] - x \frac{d}{d x} [e^{x}]}{e^{x \cdot 2}}$

Step 3.1.2

Move $2$ to the left of $x$ .

$6 \frac{e^{x} \frac{d}{d x} [x] - x \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 3.2

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

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

Step 3.3

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

$6 \frac{e^{x} - x \frac{d}{d x} [e^{x}]}{e^{2 x}}$

Step 4

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

$6 \frac{e^{x} - x e^{x}}{e^{2 x}}$

Step 5

Combine $6$ and $\frac{e^{x} - x e^{x}}{e^{2 x}}$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $- 1$ by $6$ .

$\frac{6 e^{x} - 6 x e^{x}}{e^{2 x}}$

Step 6.3

Reorder terms.

$\frac{- 6 e^{x} x + 6 e^{x}}{e^{2 x}}$

Step 6.4

Factor $6 e^{x}$ out of $- 6 e^{x} x + 6 e^{x}$ .

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Step 6.4.1

Factor $6 e^{x}$ out of $- 6 e^{x} x$ .

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

Step 6.4.2

Factor $6 e^{x}$ out of $6 e^{x}$ .

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

Step 6.4.3

Factor $6 e^{x}$ out of $6 e^{x} (- x) + 6 e^{x} (1)$ .

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

Step 6.5

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

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Step 6.5.1

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

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

Step 6.5.2

Cancel the common factors.

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Step 6.5.2.1

Multiply by $1$ .

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

Step 6.5.2.2

Cancel the common factor.

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

Step 6.5.2.3

Rewrite the expression.

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

Step 6.5.2.4

Divide $6 e^{- x} (- x + 1)$ by $1$ .

$6 e^{- x} (- x + 1)$

Step 6.6

Apply the distributive property.

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

Step 6.7

Rewrite using the commutative property of multiplication.

$6 \cdot - 1 e^{- x} x + 6 e^{- x} \cdot 1$

Step 6.8

Multiply $6$ by $1$ .

$6 \cdot - 1 e^{- x} x + 6 e^{- x}$

Step 6.9

Multiply $6$ by $- 1$ .

$- 6 e^{- x} x + 6 e^{- x}$

Step 6.10

Reorder factors in $- 6 e^{- x} x + 6 e^{- x}$ .

$- 6 x e^{- x} + 6 e^{- x}$