Calculus Examples

$y = \frac{e^{- x} + 1}{e^{x}}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\frac{e^{- x} + 1}{e^{x}})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = e^{- x} + 1$ and $g (y) = e^{x}$ .

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

Step 3.2

Differentiate using the Sum Rule.

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Step 3.2.1

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

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.2

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

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

Step 3.3

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

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Step 3.3.1

To apply the Chain Rule, set $u_{1}$ as $- x$ .

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

Step 3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

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

Step 3.3.3

Replace all occurrences of $u_{1}$ with $- x$ .

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

Step 3.4

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

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

Step 3.5

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

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

Step 3.6

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

$\frac{e^{x} (e^{- x} (- x') + 0) - (e^{- x} + 1) \frac{d}{d y} [e^{x}]}{e^{2 x}}$

Step 3.7

Add $e^{- x} (- x')$ and $0$ .

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

Step 3.8

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

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Step 3.8.1

Move $e^{- x}$ .

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

Step 3.8.2

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

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

Step 3.8.3

Add $- x$ and $x$ .

$\frac{e^{0} (- x') - (e^{- x} + 1) \frac{d}{d y} [e^{x}]}{e^{2 x}}$

Step 3.9

Simplify $e^{0} (- x')$ .

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

Step 3.10

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

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Step 3.10.1

To apply the Chain Rule, set $u_{2}$ as $x$ .

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

Step 3.10.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

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

Step 3.10.3

Replace all occurrences of $u_{2}$ with $x$ .

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

Step 3.11

Rewrite $\frac{d}{d y} [x]$ as $x'$ .

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

Step 3.12

Simplify.

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Step 3.12.1

Apply the distributive property.

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

Step 3.12.2

Apply the distributive property.

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

Step 3.12.3

Simplify the numerator.

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Step 3.12.3.1

Simplify each term.

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Step 3.12.3.1.1

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

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Step 3.12.3.1.1.1

Move $e^{x}$ .

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

Step 3.12.3.1.1.2

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

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

Step 3.12.3.1.1.3

Subtract $x$ from $x$ .

$\frac{- x' - e^{0} x' - 1 \cdot 1 (e^{x} x')}{e^{2 x}}$

Step 3.12.3.1.2

Simplify $- e^{0} x'$ .

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

Step 3.12.3.1.3

Rewrite $- 1 x'$ as $- x'$ .

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

Step 3.12.3.1.4

Multiply $- 1$ by $1$ .

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

Step 3.12.3.1.5

Rewrite $- 1 (e^{x} x')$ as $- (e^{x} x')$ .

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

Step 3.12.3.2

Subtract $x'$ from $- x'$ .

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

Step 3.12.4

Reorder terms.

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

Step 3.12.5

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

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Step 3.12.5.1

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

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

Step 3.12.5.2

Factor $x'$ out of $- 2 x'$ .

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

Step 3.12.5.3

Factor $x'$ out of $x' (- e^{x}) + x' \cdot - 2$ .

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

Step 3.12.6

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

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

Step 3.12.7

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 3.12.8

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

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

Step 3.12.9

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

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

Step 3.12.10

Move the negative in front of the fraction.

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

Step 3.12.11

Reorder factors in $- \frac{x' (e^{x} + 2)}{e^{2 x}}$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

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

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $- \frac{(e^{x} + 2) x'}{e^{2 x}} = 1$ .

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

Step 5.2

Divide each term in $- \frac{(e^{x} + 2) x'}{e^{2 x}} = 1$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- \frac{(e^{x} + 2) x'}{e^{2 x}} = 1$ by $- 1$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2

Simplify the expression.

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Step 5.2.2.2.1

Divide $\frac{(e^{x} + 2) x'}{e^{2 x}}$ by $1$ .

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

Step 5.2.2.2.2

Reorder factors in $\frac{(e^{x} + 2) x'}{e^{2 x}}$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $1$ by $- 1$ .

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

Step 5.3

Multiply both sides by $e^{2 x}$ .

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

Step 5.4

Simplify the left side.

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Step 5.4.1

Simplify $\frac{x' (e^{x} + 2)}{e^{2 x}} e^{2 x}$ .

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Step 5.4.1.1

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

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Step 5.4.1.1.1

Cancel the common factor.

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

Step 5.4.1.1.2

Rewrite the expression.

$x' (e^{x} + 2) = - e^{2 x}$

Step 5.4.1.2

Apply the distributive property.

$x' e^{x} + x' \cdot 2 = - e^{2 x}$

Step 5.4.1.3

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

$x' e^{x} + 2 x' = - e^{2 x}$

Step 5.5

Solve for $x'$ .

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Step 5.5.1

Factor $x'$ out of $x' e^{x} + 2 x'$ .

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Step 5.5.1.1

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

$x' (e^{x}) + 2 x' = - e^{2 x}$

Step 5.5.1.2

Factor $x'$ out of $2 x'$ .

$x' (e^{x}) + x' \cdot 2 = - e^{2 x}$

Step 5.5.1.3

Factor $x'$ out of $x' (e^{x}) + x' \cdot 2$ .

$x' (e^{x} + 2) = - e^{2 x}$

Step 5.5.2

Divide each term in $x' (e^{x} + 2) = - e^{2 x}$ by $e^{x} + 2$ and simplify.

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Step 5.5.2.1

Divide each term in $x' (e^{x} + 2) = - e^{2 x}$ by $e^{x} + 2$ .

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

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

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

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Step 5.5.2.2.1.1

Cancel the common factor.

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

Step 5.5.2.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{- e^{2 x}}{e^{x} + 2}$

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Move the negative in front of the fraction.

$x' = - \frac{e^{2 x}}{e^{x} + 2}$

Step 6

Replace $x'$ with $\frac{d x}{d y}$ .

$\frac{d x}{d y} = - \frac{e^{2 x}}{e^{x} + 2}$