Find the Derivative minus d First, divide dx (three minus xe to the power of x) divide(x plus e to the power of x)

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

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

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

Differentiate.

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By the Sum Rule, the derivative of $3 - x e^{x}$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- x e^{x}]$ .

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

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{(x + e^{x}) (0 + \frac{d}{d x} [- x e^{x}]) - (3 - x e^{x}) \frac{d}{d x} [x + e^{x}]}{{(x + e^{x})}^{2}}$

Sum $0$ and $\frac{d}{d x} [- x e^{x}]$ .

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

Since $- 1$ is constant with respect to $x$ , the derivative of $- x e^{x}$ with respect to $x$ is $- \frac{d}{d x} [x e^{x}]$ .

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

Clarify the expression.

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Move $- 1$ to the left of $x + e^{x}$ .

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

Should be rewritten $- 1 (x + e^{x})$ as $- (x + e^{x})$ .

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

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{x}$ .

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

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

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

Differentiate.

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

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

First, multiply $e^{x}$ by $1$ .

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

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

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

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

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

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

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

Simplify.

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Use the distributive law.

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

Apply the distributive property.

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

Clarify the numerator.

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Clarify each term.

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Expand $(- x - e^{x}) (x e^{x} + e^{x})$ using the FOIL Method.

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Clarify each term.

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Multiply $x$ by $x$ by adding the exponents.

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Put $x$ .

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

Multiply $x$ by $x$ .

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

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

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Move $e^{x}$ .

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

Apply the power rule $a^{m} a^{n} = a^{m + n}$ to Add exponents.

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

Add $x$ and $x$ .

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

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

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Move $e^{x}$ .

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

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

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

Add $x$ and $x$ .

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

Simplify each term.

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Multiply $- 1$ by $3$ .

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

Multiply $- (- x e^{x})$ .

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Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

Expand $(- 3 + x e^{x}) (1 + e^{x})$ using the FOIL Method.

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

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Multiply $- 3$ by $1$ .

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

Multiply $x$ by $1$ .

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

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

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Move $e^{x}$ .

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

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

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

Add $x$ and $x$ .

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

Combine the opposite terms in $- x^{2} e^{x} - x e^{x} - e^{2 x} x - e^{2 x} - 3 - 3 e^{x} + x e^{x} + x e^{2 x}$ .

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Add $- x e^{x}$ and $x e^{x}$ .

$\frac{- x^{2} e^{x} + 0 - e^{2 x} x - e^{2 x} - 3 - 3 e^{x} + x e^{2 x}}{{(x + e^{x})}^{2}}$

Add $- x^{2} e^{x}$ and $0$ .

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

Shift the factors in the terms $- e^{2 x} x$ and $x e^{2 x}$ .

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

Add $- e^{2 x} x$ and $e^{2 x} x$ .

$\frac{- x^{2} e^{x} + 0 - e^{2 x} - 3 - 3 e^{x}}{{(x + e^{x})}^{2}}$

Add $- x^{2} e^{x}$ and $0$ .

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

Factor $- 1$ out of $- x^{2} e^{x}$ .

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

Factor $- 1$ out of $- e^{2 x}$ .

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

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

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

Rewrite $- 3$ as $- 1 (3)$ .

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

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

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

Factor $- 1$ out of $- 3 e^{x}$ .

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

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

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

Rewrite $- (x^{2} e^{x} + e^{2 x} + 3 + 3 e^{x})$ as $- 1 (x^{2} e^{x} + e^{2 x} + 3 + 3 e^{x})$ .

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

Put the negative in front of the fraction.

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

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