Calculus Examples

$\frac{d P}{d t} = P (1 - P)$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{P (1 - P)}$ .

$\frac{1}{P (1 - P)} \frac{d P}{d t} = \frac{1}{P (1 - P)} (P (1 - P))$

Step 1.2

Cancel the common factor of $P (1 - P)$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{P (1 - P)} \frac{d P}{d t} = \frac{1}{P (1 - P)} (P (1 - P))$

Step 1.2.2

Rewrite the expression.

$\frac{1}{P (1 - P)} \frac{d P}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{P (1 - P)} d P = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{P (1 - P)} d P = \int d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Write the fraction using partial fraction decomposition.

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Step 2.2.1.1

Decompose the fraction and multiply through by the common denominator.

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Step 2.2.1.1.1

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $B$ .

$\frac{A}{P} + \frac{B}{1 - P}$

Step 2.2.1.1.2

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $P (1 - P)$ .

$\frac{1 (P (1 - P))}{P (1 - P)} = \frac{A (P (1 - P))}{P} + \frac{(B) (P (1 - P))}{1 - P}$

Step 2.2.1.1.3

Cancel the common factor of $P$ .

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Step 2.2.1.1.3.1

Cancel the common factor.

$\frac{1 (P (1 - P))}{P (1 - P)} = \frac{A (P (1 - P))}{P} + \frac{(B) (P (1 - P))}{1 - P}$

Step 2.2.1.1.3.2

Rewrite the expression.

$\frac{1 (1 - P)}{1 - P} = \frac{A (P (1 - P))}{P} + \frac{(B) (P (1 - P))}{1 - P}$

Step 2.2.1.1.4

Cancel the common factor of $1 - P$ .

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Step 2.2.1.1.4.1

Cancel the common factor.

$\frac{1 (1 - P)}{1 - P} = \frac{A (P (1 - P))}{P} + \frac{(B) (P (1 - P))}{1 - P}$

Step 2.2.1.1.4.2

Rewrite the expression.

$1 = \frac{A (P (1 - P))}{P} + \frac{(B) (P (1 - P))}{1 - P}$

Step 2.2.1.1.5

Simplify each term.

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Step 2.2.1.1.5.1

Cancel the common factor of $P$ .

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Step 2.2.1.1.5.1.1

Cancel the common factor.

$1 = \frac{A (P (1 - P))}{P} + \frac{(B) (P (1 - P))}{1 - P}$

Step 2.2.1.1.5.1.2

Divide $A (1 - P)$ by $1$ .

$1 = A (1 - P) + \frac{(B) (P (1 - P))}{1 - P}$

Step 2.2.1.1.5.2

Apply the distributive property.

$1 = A \cdot 1 + A (- P) + \frac{(B) (P (1 - P))}{1 - P}$

Step 2.2.1.1.5.3

Multiply $A$ by $1$ .

$1 = A + A (- P) + \frac{(B) (P (1 - P))}{1 - P}$

Step 2.2.1.1.5.4

Rewrite using the commutative property of multiplication.

$1 = A - A P + \frac{(B) (P (1 - P))}{1 - P}$

Step 2.2.1.1.5.5

Cancel the common factor of $1 - P$ .

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Step 2.2.1.1.5.5.1

Cancel the common factor.

$1 = A - A P + \frac{B (P (1 - P))}{1 - P}$

Step 2.2.1.1.5.5.2

Divide $(B) (P)$ by $1$ .

$1 = A - A P + (B) (P)$

$1 = A - A P + B P$

Step 2.2.1.1.6

Move $A$ .

$1 = - A P + B P + A$

Step 2.2.1.2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.2.1.2.1

Create an equation for the partial fraction variables by equating the coefficients of $P$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = - 1 A + B$

Step 2.2.1.2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $P$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = A$

Step 2.2.1.2.3

Set up the system of equations to find the coefficients of the partial fractions.

$0 = - 1 A + B$

$1 = A$

$0 = - 1 A + B$

$1 = A$

Step 2.2.1.3

Solve the system of equations.

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Step 2.2.1.3.1

Rewrite the equation as $A = 1$ .

$A = 1$

$0 = - 1 A + B$

Step 2.2.1.3.2

Replace all occurrences of $A$ with $1$ in each equation.

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Step 2.2.1.3.2.1

Replace all occurrences of $A$ in $0 = - 1 A + B$ with $1$ .

$0 = - 1 \cdot 1 + B$

$A = 1$

Step 2.2.1.3.2.2

Simplify the right side.

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Step 2.2.1.3.2.2.1

Multiply $- 1$ by $1$ .

$0 = - 1 + B$

$A = 1$

$0 = - 1 + B$

$A = 1$

$0 = - 1 + B$

$A = 1$

Step 2.2.1.3.3

Solve for $B$ in $0 = - 1 + B$ .

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Step 2.2.1.3.3.1

Rewrite the equation as $- 1 + B = 0$ .

$- 1 + B = 0$

$A = 1$

Step 2.2.1.3.3.2

Add $1$ to both sides of the equation.

$B = 1$

$A = 1$

$B = 1$

$A = 1$

Step 2.2.1.3.4

Solve the system of equations.

$B = 1 A = 1$

Step 2.2.1.3.5

List all of the solutions.

$B = 1, A = 1$

Step 2.2.1.4

Replace each of the partial fraction coefficients in $\frac{A}{P} + \frac{B}{1 - P}$ with the values found for $A$ and $B$ .

$\frac{1}{P} + \frac{1}{1 - P}$

Step 2.2.1.5

Remove the zero from the expression.

$\int \frac{1}{P} + \frac{1}{1 - P} d P = \int d t$

Step 2.2.2

Split the single integral into multiple integrals.

$\int \frac{1}{P} d P + \int \frac{1}{1 - P} d P = \int d t$

Step 2.2.3

The integral of $\frac{1}{P}$ with respect to $P$ is $\ln (| P |)$ .

$\ln (| P |) + C_{1} + \int \frac{1}{1 - P} d P = \int d t$

Step 2.2.4

Let $u = 1 - P$ . Then $d u = - d P$ , so $- d u = d P$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.4.1

Let $u = 1 - P$ . Find $\frac{d u}{d P}$ .

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Step 2.2.4.1.1

Rewrite.

$\frac{- 1}{1}$

Step 2.2.4.1.2

Divide $- 1$ by $1$ .

$- 1$

Step 2.2.4.2

Rewrite the problem using $u$ and $d u$ .

$\ln (| P |) + C_{1} + \int \frac{- 1}{u} d u = \int d t$

Step 2.2.5

Move the negative in front of the fraction.

$\ln (| P |) + C_{1} + \int - \frac{1}{u} d u = \int d t$

Step 2.2.6

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\ln (| P |) + C_{1} - \int \frac{1}{u} d u = \int d t$

Step 2.2.7

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\ln (| P |) + C_{1} - (\ln (| u |) + C_{2}) = \int d t$

Step 2.2.8

Simplify.

$\ln (| P |) - \ln (| u |) + C_{3} = \int d t$

Step 2.2.9

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| P |}{| u |}) + C_{3} = \int d t$

Step 2.2.10

Replace all occurrences of $u$ with $1 - P$ .

$\ln (\frac{| P |}{| 1 - P |}) + C_{3} = \int d t$

Step 2.3

Apply the constant rule.

$\ln (\frac{| P |}{| 1 - P |}) + C_{3} = t + C_{4}$

Step 2.4

Group the constant of integration on the right side as $C$ .

$\ln (\frac{| P |}{| 1 - P |}) = t + C$

Step 3

Solve for $P$ .

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Step 3.1

To solve for $P$ , rewrite the equation using properties of logarithms.

$e^{\ln (\frac{| P |}{| 1 - P |})} = e^{t + C}$

Step 3.2

Rewrite $\ln (\frac{| P |}{| 1 - P |}) = t + C$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{t + C} = \frac{| P |}{| 1 - P |}$

Step 3.3

Solve for $P$ .

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Step 3.3.1

Rewrite the equation as $\frac{| P |}{| 1 - P |} = e^{t + C}$ .

$\frac{| P |}{| 1 - P |} = e^{t + C}$

Step 3.3.2

Multiply both sides by $| 1 - P |$ .

$\frac{| P |}{| 1 - P |} | 1 - P | = e^{t + C} | 1 - P |$

Step 3.3.3

Simplify the left side.

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Step 3.3.3.1

Cancel the common factor of $| 1 - P |$ .

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Step 3.3.3.1.1

Cancel the common factor.

$\frac{| P |}{| 1 - P |} | 1 - P | = e^{t + C} | 1 - P |$

Step 3.3.3.1.2

Rewrite the expression.

$| P | = e^{t + C} | 1 - P |$

Step 3.3.4

Solve for $P$ .

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Step 3.3.4.1

Rewrite the equation as $e^{t + C} | 1 - P | = | P |$ .

$e^{t + C} | 1 - P | = | P |$

Step 3.3.4.2

Divide each term in $e^{t + C} | 1 - P | = | P |$ by $e^{t + C}$ and simplify.

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Step 3.3.4.2.1

Divide each term in $e^{t + C} | 1 - P | = | P |$ by $e^{t + C}$ .

$\frac{e^{t + C} | 1 - P |}{e^{t + C}} = \frac{| P |}{e^{t + C}}$

Step 3.3.4.2.2

Simplify the left side.

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Step 3.3.4.2.2.1

Cancel the common factor of $e^{t + C}$ .

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Step 3.3.4.2.2.1.1

Cancel the common factor.

$\frac{e^{t + C} | 1 - P |}{e^{t + C}} = \frac{| P |}{e^{t + C}}$

Step 3.3.4.2.2.1.2

Divide $| 1 - P |$ by $1$ .

$| 1 - P | = \frac{| P |}{e^{t + C}}$

Step 3.3.4.3

Rewrite the absolute value equation as four equations without absolute value bars.

$1 - P = \frac{P}{e^{t + C}}$

$1 - P = - \frac{P}{e^{t + C}}$

$- (1 - P) = \frac{P}{e^{t + C}}$

$- (1 - P) = - \frac{P}{e^{t + C}}$

Step 3.3.4.4

After simplifying, there are only two unique equations to be solved.

$1 - P = \frac{P}{e^{t + C}}$

$1 - P = - \frac{P}{e^{t + C}}$

Step 3.3.4.5

Solve $1 - P = \frac{P}{e^{t + C}}$ for $P$ .

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Step 3.3.4.5.1

Multiply both sides by $e^{t + C}$ .

$(1 - P) e^{t + C} = \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2

Simplify.

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Step 3.3.4.5.2.1

Simplify the left side.

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Step 3.3.4.5.2.1.1

Simplify $(1 - P) e^{t + C}$ .

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Step 3.3.4.5.2.1.1.1

Apply the distributive property.

$1 e^{t + C} - P e^{t + C} = \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2.1.1.2

Simplify the expression.

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Step 3.3.4.5.2.1.1.2.1

Multiply $e^{t + C}$ by $1$ .

$e^{t + C} - P e^{t + C} = \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2.1.1.2.2

Reorder $t$ and $C$ .

$e^{C + t} - P e^{t + C} = \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2.1.1.2.3

Reorder $t$ and $C$ .

$e^{C + t} - P e^{C + t} = \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2.1.1.2.4

Reorder $e^{C + t}$ and $- P e^{C + t}$ .

$- P e^{C + t} + e^{C + t} = \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2.2

Simplify the right side.

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Step 3.3.4.5.2.2.1

Cancel the common factor of $e^{t + C}$ .

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Step 3.3.4.5.2.2.1.1

Cancel the common factor.

$- P e^{C + t} + e^{C + t} = \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.5.2.2.1.2

Rewrite the expression.

$- P e^{C + t} + e^{C + t} = P$

Step 3.3.4.5.3

Solve for $P$ .

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Step 3.3.4.5.3.1

Subtract $P$ from both sides of the equation.

$- P e^{C + t} + e^{C + t} - P = 0$

Step 3.3.4.5.3.2

Subtract $e^{C + t}$ from both sides of the equation.

$- P e^{C + t} - P = - e^{C + t}$

Step 3.3.4.5.3.3

Factor $P$ out of $- P e^{C + t} - P$ .

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Step 3.3.4.5.3.3.1

Factor $P$ out of $- P e^{C + t}$ .

$P (- 1 e^{C + t}) - P = - e^{C + t}$

Step 3.3.4.5.3.3.2

Factor $P$ out of $- P$ .

$P (- 1 e^{C + t}) + P \cdot - 1 = - e^{C + t}$

Step 3.3.4.5.3.3.3

Factor $P$ out of $P (- 1 e^{C + t}) + P \cdot - 1$ .

$P (- 1 e^{C + t} - 1) = - e^{C + t}$

Step 3.3.4.5.3.4

Rewrite $- 1 e^{C + t}$ as $- e^{C + t}$ .

$P (- e^{C + t} - 1) = - e^{C + t}$

Step 3.3.4.5.3.5

Divide each term in $P (- e^{C + t} - 1) = - e^{C + t}$ by $- e^{C + t} - 1$ and simplify.

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Step 3.3.4.5.3.5.1

Divide each term in $P (- e^{C + t} - 1) = - e^{C + t}$ by $- e^{C + t} - 1$ .

$\frac{P (- e^{C + t} - 1)}{- e^{C + t} - 1} = \frac{- e^{C + t}}{- e^{C + t} - 1}$

Step 3.3.4.5.3.5.2

Simplify the left side.

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Step 3.3.4.5.3.5.2.1

Cancel the common factor of $- e^{C + t} - 1$ .

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Step 3.3.4.5.3.5.2.1.1

Cancel the common factor.

$\frac{P (- e^{C + t} - 1)}{- e^{C + t} - 1} = \frac{- e^{C + t}}{- e^{C + t} - 1}$

Step 3.3.4.5.3.5.2.1.2

Divide $P$ by $1$ .

$P = \frac{- e^{C + t}}{- e^{C + t} - 1}$

Step 3.3.4.5.3.5.3

Simplify the right side.

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Step 3.3.4.5.3.5.3.1

Move the negative in front of the fraction.

$P = - \frac{e^{C + t}}{- e^{C + t} - 1}$

Step 3.3.4.5.3.5.3.2

Factor $- 1$ out of $- e^{C + t}$ .

$P = - \frac{e^{C + t}}{- e^{C + t} - 1}$

Step 3.3.4.5.3.5.3.3

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

$P = - \frac{e^{C + t}}{- e^{C + t} - 1 (1)}$

Step 3.3.4.5.3.5.3.4

Factor $- 1$ out of $- e^{C + t} - 1 (1)$ .

$P = - \frac{e^{C + t}}{- (e^{C + t} + 1)}$

Step 3.3.4.5.3.5.3.5

Simplify the expression.

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Step 3.3.4.5.3.5.3.5.1

Rewrite $- (e^{C + t} + 1)$ as $- 1 (e^{C + t} + 1)$ .

$P = - \frac{e^{C + t}}{- 1 (e^{C + t} + 1)}$

Step 3.3.4.5.3.5.3.5.2

Move the negative in front of the fraction.

$P = - - \frac{e^{C + t}}{e^{C + t} + 1}$

Step 3.3.4.5.3.5.3.5.3

Multiply $- 1$ by $- 1$ .

$P = 1 \frac{e^{C + t}}{e^{C + t} + 1}$

Step 3.3.4.5.3.5.3.5.4

Multiply $\frac{e^{C + t}}{e^{C + t} + 1}$ by $1$ .

$P = \frac{e^{C + t}}{e^{C + t} + 1}$

Step 3.3.4.6

Solve $1 - P = - \frac{P}{e^{t + C}}$ for $P$ .

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Step 3.3.4.6.1

Multiply both sides by $e^{t + C}$ .

$(1 - P) e^{t + C} = - \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2

Simplify.

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Step 3.3.4.6.2.1

Simplify the left side.

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Step 3.3.4.6.2.1.1

Simplify $(1 - P) e^{t + C}$ .

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Step 3.3.4.6.2.1.1.1

Apply the distributive property.

$1 e^{t + C} - P e^{t + C} = - \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.1.1.2

Simplify the expression.

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Step 3.3.4.6.2.1.1.2.1

Multiply $e^{t + C}$ by $1$ .

$e^{t + C} - P e^{t + C} = - \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.1.1.2.2

Reorder $t$ and $C$ .

$e^{C + t} - P e^{t + C} = - \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.1.1.2.3

Reorder $t$ and $C$ .

$e^{C + t} - P e^{C + t} = - \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.1.1.2.4

Reorder $e^{C + t}$ and $- P e^{C + t}$ .

$- P e^{C + t} + e^{C + t} = - \frac{P}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.2

Simplify the right side.

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Step 3.3.4.6.2.2.1

Cancel the common factor of $e^{t + C}$ .

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Step 3.3.4.6.2.2.1.1

Move the leading negative in $- \frac{P}{e^{t + C}}$ into the numerator.

$- P e^{C + t} + e^{C + t} = \frac{- P}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.2.1.2

Cancel the common factor.

$- P e^{C + t} + e^{C + t} = \frac{- P}{e^{t + C}} e^{t + C}$

Step 3.3.4.6.2.2.1.3

Rewrite the expression.

$- P e^{C + t} + e^{C + t} = - P$

Step 3.3.4.6.3

Solve for $P$ .

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Step 3.3.4.6.3.1

Add $P$ to both sides of the equation.

$- P e^{C + t} + e^{C + t} + P = 0$

Step 3.3.4.6.3.2

Subtract $e^{C + t}$ from both sides of the equation.

$- P e^{C + t} + P = - e^{C + t}$

Step 3.3.4.6.3.3

Factor $P$ out of $- P e^{C + t} + P$ .

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Step 3.3.4.6.3.3.1

Factor $P$ out of $- P e^{C + t}$ .

$P (- 1 e^{C + t}) + P = - e^{C + t}$

Step 3.3.4.6.3.3.2

Raise $P$ to the power of $1$ .

$P (- 1 e^{C + t}) + P = - e^{C + t}$

Step 3.3.4.6.3.3.3

Factor $P$ out of $P^{1}$ .

$P (- 1 e^{C + t}) + P \cdot 1 = - e^{C + t}$

Step 3.3.4.6.3.3.4

Factor $P$ out of $P (- 1 e^{C + t}) + P \cdot 1$ .

$P (- 1 e^{C + t} + 1) = - e^{C + t}$

Step 3.3.4.6.3.4

Rewrite $- 1 e^{C + t}$ as $- e^{C + t}$ .

$P (- e^{C + t} + 1) = - e^{C + t}$

Step 3.3.4.6.3.5

Divide each term in $P (- e^{C + t} + 1) = - e^{C + t}$ by $- e^{C + t} + 1$ and simplify.

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Step 3.3.4.6.3.5.1

Divide each term in $P (- e^{C + t} + 1) = - e^{C + t}$ by $- e^{C + t} + 1$ .

$\frac{P (- e^{C + t} + 1)}{- e^{C + t} + 1} = \frac{- e^{C + t}}{- e^{C + t} + 1}$

Step 3.3.4.6.3.5.2

Simplify the left side.

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Step 3.3.4.6.3.5.2.1

Cancel the common factor of $- e^{C + t} + 1$ .

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Step 3.3.4.6.3.5.2.1.1

Cancel the common factor.

$\frac{P (- e^{C + t} + 1)}{- e^{C + t} + 1} = \frac{- e^{C + t}}{- e^{C + t} + 1}$

Step 3.3.4.6.3.5.2.1.2

Divide $P$ by $1$ .

$P = \frac{- e^{C + t}}{- e^{C + t} + 1}$

Step 3.3.4.6.3.5.3

Simplify the right side.

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Step 3.3.4.6.3.5.3.1

Move the negative in front of the fraction.

$P = - \frac{e^{C + t}}{- e^{C + t} + 1}$

Step 3.3.4.6.3.5.3.2

Factor $- 1$ out of $- e^{C + t}$ .

$P = - \frac{e^{C + t}}{- e^{C + t} + 1}$

Step 3.3.4.6.3.5.3.3

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

$P = - \frac{e^{C + t}}{- e^{C + t} - 1 (- 1)}$

Step 3.3.4.6.3.5.3.4

Factor $- 1$ out of $- e^{C + t} - 1 (- 1)$ .

$P = - \frac{e^{C + t}}{- (e^{C + t} - 1)}$

Step 3.3.4.6.3.5.3.5

Simplify the expression.

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Step 3.3.4.6.3.5.3.5.1

Rewrite $- (e^{C + t} - 1)$ as $- 1 (e^{C + t} - 1)$ .

$P = - \frac{e^{C + t}}{- 1 (e^{C + t} - 1)}$

Step 3.3.4.6.3.5.3.5.2

Move the negative in front of the fraction.

$P = - - \frac{e^{C + t}}{e^{C + t} - 1}$

Step 3.3.4.6.3.5.3.5.3

Multiply $- 1$ by $- 1$ .

$P = 1 \frac{e^{C + t}}{e^{C + t} - 1}$

Step 3.3.4.6.3.5.3.5.4

Multiply $\frac{e^{C + t}}{e^{C + t} - 1}$ by $1$ .

$P = \frac{e^{C + t}}{e^{C + t} - 1}$

Step 3.3.4.7

List all of the solutions.

$P = \frac{e^{C + t}}{e^{C + t} + 1}$

$P = \frac{e^{C + t}}{e^{C + t} - 1}$

$P = \frac{e^{C + t}}{e^{C + t} + 1}$

$P = \frac{e^{C + t}}{e^{C + t} - 1}$

$P = \frac{e^{C + t}}{e^{C + t} + 1}$

$P = \frac{e^{C + t}}{e^{C + t} - 1}$

$P = \frac{e^{C + t}}{e^{C + t} + 1}$

$P = \frac{e^{C + t}}{e^{C + t} - 1}$

Step 4

Group the constant terms together.

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Step 4.1

Reorder terms.

$P = \frac{e^{t + C}}{e^{C + t} + 1}$

$P = \frac{e^{C + t}}{e^{C + t} - 1}$

Step 4.2

Rewrite $e^{t + C}$ as $e^{t} e^{C}$ .

$P = \frac{e^{t} e^{C}}{e^{C + t} + 1}$

$P = \frac{e^{C + t}}{e^{C + t} - 1}$

Step 4.3

Reorder $e^{t}$ and $e^{C}$ .

$P = \frac{e^{C} e^{t}}{e^{C + t} + 1}$

$P = \frac{e^{C + t}}{e^{C + t} - 1}$

Step 4.4

Reorder terms.

$P = \frac{e^{C} e^{t}}{e^{t + C} + 1}$

$P = \frac{e^{C + t}}{e^{C + t} - 1}$

Step 4.5

Rewrite $e^{t + C}$ as $e^{t} e^{C}$ .

$P = \frac{e^{C} e^{t}}{e^{t} e^{C} + 1}$

$P = \frac{e^{C + t}}{e^{C + t} - 1}$

Step 4.6

Reorder $e^{t}$ and $e^{C}$ .

$P = \frac{e^{C} e^{t}}{e^{C} e^{t} + 1}$

$P = \frac{e^{C + t}}{e^{C + t} - 1}$

Step 4.7

Reorder terms.

$P = \frac{e^{C} e^{t}}{e^{C} e^{t} + 1}$

$P = \frac{e^{t + C}}{e^{C + t} - 1}$

Step 4.8

Rewrite $e^{t + C}$ as $e^{t} e^{C}$ .

$P = \frac{e^{C} e^{t}}{e^{C} e^{t} + 1}$

$P = \frac{e^{t} e^{C}}{e^{C + t} - 1}$

Step 4.9

Reorder $e^{t}$ and $e^{C}$ .

$P = \frac{e^{C} e^{t}}{e^{C} e^{t} + 1}$

$P = \frac{e^{C} e^{t}}{e^{C + t} - 1}$

Step 4.10

Reorder terms.

$P = \frac{e^{C} e^{t}}{e^{C} e^{t} + 1}$

$P = \frac{e^{C} e^{t}}{e^{t + C} - 1}$

Step 4.11

Rewrite $e^{t + C}$ as $e^{t} e^{C}$ .

$P = \frac{e^{C} e^{t}}{e^{C} e^{t} + 1}$

$P = \frac{e^{C} e^{t}}{e^{t} e^{C} - 1}$

Step 4.12

Reorder $e^{t}$ and $e^{C}$ .

$P = \frac{e^{C} e^{t}}{e^{C} e^{t} + 1}$

$P = \frac{e^{C} e^{t}}{e^{C} e^{t} - 1}$

$P = \frac{e^{C} e^{t}}{e^{C} e^{t} + 1}$

$P = \frac{e^{C} e^{t}}{e^{C} e^{t} - 1}$