Calculus Examples

$\frac{A}{1 + B e^{\frac{t}{2}}} = C$

Step 1

Multiply both sides by $1 + B e^{\frac{t}{2}}$ .

$\frac{A}{1 + B e^{\frac{t}{2}}} (1 + B e^{\frac{t}{2}}) = C (1 + B e^{\frac{t}{2}})$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $1 + B e^{\frac{t}{2}}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{A}{1 + B e^{\frac{t}{2}}} (1 + B e^{\frac{t}{2}}) = C (1 + B e^{\frac{t}{2}})$

Step 2.1.1.2

Rewrite the expression.

$A = C (1 + B e^{\frac{t}{2}})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $C (1 + B e^{\frac{t}{2}})$ .

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Step 2.2.1.1

Apply the distributive property.

$A = C \cdot 1 + C (B e^{\frac{t}{2}})$

Step 2.2.1.2

Simplify the expression.

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Step 2.2.1.2.1

Multiply $C$ by $1$ .

$A = C + C (B e^{\frac{t}{2}})$

Step 2.2.1.2.2

Reorder $C$ and $C B e^{\frac{t}{2}}$ .

$A = C B e^{\frac{t}{2}} + C$

Step 3

Solve for $t$ .

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Step 3.1

Rewrite the equation as $C B e^{\frac{t}{2}} + C = A$ .

$C B e^{\frac{t}{2}} + C = A$

Step 3.2

Subtract $C$ from both sides of the equation.

$C B e^{\frac{t}{2}} = A - C$

Step 3.3

Divide each term in $C B e^{\frac{t}{2}} = A - C$ by $C B$ and simplify.

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Step 3.3.1

Divide each term in $C B e^{\frac{t}{2}} = A - C$ by $C B$ .

$\frac{C B e^{\frac{t}{2}}}{C B} = \frac{A}{C B} + \frac{- C}{C B}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $C$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{C B e^{\frac{t}{2}}}{C B} = \frac{A}{C B} + \frac{- C}{C B}$

Step 3.3.2.1.2

Rewrite the expression.

$\frac{B e^{\frac{t}{2}}}{B} = \frac{A}{C B} + \frac{- C}{C B}$

Step 3.3.2.2

Cancel the common factor of $B$ .

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Step 3.3.2.2.1

Cancel the common factor.

$\frac{B e^{\frac{t}{2}}}{B} = \frac{A}{C B} + \frac{- C}{C B}$

Step 3.3.2.2.2

Divide $e^{\frac{t}{2}}$ by $1$ .

$e^{\frac{t}{2}} = \frac{A}{C B} + \frac{- C}{C B}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Cancel the common factor of $C$ .

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Step 3.3.3.1.1.1

Cancel the common factor.

$e^{\frac{t}{2}} = \frac{A}{C B} + \frac{- C}{C B}$

Step 3.3.3.1.1.2

Rewrite the expression.

$e^{\frac{t}{2}} = \frac{A}{C B} + \frac{- 1}{B}$

Step 3.3.3.1.2

Move the negative in front of the fraction.

$e^{\frac{t}{2}} = \frac{A}{C B} - \frac{1}{B}$

Step 3.4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{\frac{t}{2}}) = \ln (\frac{A}{C B} - \frac{1}{B})$

Step 3.5

Expand the left side.

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Step 3.5.1

Expand $\ln (e^{\frac{t}{2}})$ by moving $\frac{t}{2}$ outside the logarithm.

$\frac{t}{2} \ln (e) = \ln (\frac{A}{C B} - \frac{1}{B})$

Step 3.5.2

The natural logarithm of $e$ is $1$ .

$\frac{t}{2} \cdot 1 = \ln (\frac{A}{C B} - \frac{1}{B})$

Step 3.5.3

Multiply $\frac{t}{2}$ by $1$ .

$\frac{t}{2} = \ln (\frac{A}{C B} - \frac{1}{B})$

Step 3.6

Multiply both sides of the equation by $2$ .

$2 \frac{t}{2} = 2 \ln (\frac{A}{C B} - \frac{1}{B})$

Step 3.7

Simplify the left side.

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Step 3.7.1

Cancel the common factor of $2$ .

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Step 3.7.1.1

Cancel the common factor.

$2 \frac{t}{2} = 2 \ln (\frac{A}{C B} - \frac{1}{B})$

Step 3.7.1.2

Rewrite the expression.

$t = 2 \ln (\frac{A}{C B} - \frac{1}{B})$