Calculus Examples

$y = \frac{m}{1 + c e^{- km t}}$

Step 1

Rewrite the equation as $\frac{m}{1 + c e^{- k m t}} = y$ .

$\frac{m}{1 + c e^{- k m t}} = y$

Step 2

Multiply both sides by $1 + c e^{- k m t}$ .

$\frac{m}{1 + c e^{- k m t}} (1 + c e^{- k m t}) = y (1 + c e^{- k m t})$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $1 + c e^{- k m t}$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{m}{1 + c e^{- k m t}} (1 + c e^{- k m t}) = y (1 + c e^{- k m t})$

Step 3.1.1.2

Rewrite the expression.

$m = y (1 + c e^{- k m t})$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $y (1 + c e^{- k m t})$ .

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Step 3.2.1.1

Apply the distributive property.

$m = y \cdot 1 + y (c e^{- k m t})$

Step 3.2.1.2

Simplify the expression.

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Step 3.2.1.2.1

Multiply $y$ by $1$ .

$m = y + y (c e^{- k m t})$

Step 3.2.1.2.2

Reorder $y$ and $c$ .

$m = y + c y e^{- k m t}$

Step 3.2.1.2.3

Reorder $y$ and $c y e^{- k m t}$ .

$m = c y e^{- k m t} + y$

Step 4

Solve for $t$ .

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Step 4.1

Rewrite the equation as $c y e^{- k m t} + y = m$ .

$c y e^{- k m t} + y = m$

Step 4.2

Subtract $y$ from both sides of the equation.

$c y e^{- k m t} = m - y$

Step 4.3

Divide each term in $c y e^{- k m t} = m - y$ by $c y$ and simplify.

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Step 4.3.1

Divide each term in $c y e^{- k m t} = m - y$ by $c y$ .

$\frac{c y e^{- k m t}}{c y} = \frac{m}{c y} + \frac{- y}{c y}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $c$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{c y e^{- k m t}}{c y} = \frac{m}{c y} + \frac{- y}{c y}$

Step 4.3.2.1.2

Rewrite the expression.

$\frac{y e^{- k m t}}{y} = \frac{m}{c y} + \frac{- y}{c y}$

Step 4.3.2.2

Cancel the common factor of $y$ .

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Step 4.3.2.2.1

Cancel the common factor.

$\frac{y e^{- k m t}}{y} = \frac{m}{c y} + \frac{- y}{c y}$

Step 4.3.2.2.2

Divide $e^{- k m t}$ by $1$ .

$e^{- k m t} = \frac{m}{c y} + \frac{- y}{c y}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Simplify each term.

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Step 4.3.3.1.1

Cancel the common factor of $y$ .

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Step 4.3.3.1.1.1

Cancel the common factor.

$e^{- k m t} = \frac{m}{c y} + \frac{- y}{c y}$

Step 4.3.3.1.1.2

Rewrite the expression.

$e^{- k m t} = \frac{m}{c y} + \frac{- 1}{c}$

Step 4.3.3.1.2

Move the negative in front of the fraction.

$e^{- k m t} = \frac{m}{c y} - \frac{1}{c}$

Step 4.4

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

$\ln (e^{- k m t}) = \ln (\frac{m}{c y} - \frac{1}{c})$

Step 4.5

Expand the left side.

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Step 4.5.1

Expand $\ln (e^{- k m t})$ by moving $- k m t$ outside the logarithm.

$- k m t \ln (e) = \ln (\frac{m}{c y} - \frac{1}{c})$

Step 4.5.2

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

$- k m t \cdot 1 = \ln (\frac{m}{c y} - \frac{1}{c})$

Step 4.5.3

Multiply $- 1$ by $1$ .

$- k m t = \ln (\frac{m}{c y} - \frac{1}{c})$

Step 4.6

Divide each term in $- k m t = \ln (\frac{m}{c y} - \frac{1}{c})$ by $- k m$ and simplify.

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Step 4.6.1

Divide each term in $- k m t = \ln (\frac{m}{c y} - \frac{1}{c})$ by $- k m$ .

$\frac{- k m t}{- k m} = \frac{\ln (\frac{m}{c y} - \frac{1}{c})}{- k m}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Dividing two negative values results in a positive value.

$\frac{k m t}{k m} = \frac{\ln (\frac{m}{c y} - \frac{1}{c})}{- k m}$

Step 4.6.2.2

Cancel the common factor of $k$ .

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Step 4.6.2.2.1

Cancel the common factor.

$\frac{k m t}{k m} = \frac{\ln (\frac{m}{c y} - \frac{1}{c})}{- k m}$

Step 4.6.2.2.2

Rewrite the expression.

$\frac{m t}{m} = \frac{\ln (\frac{m}{c y} - \frac{1}{c})}{- k m}$

Step 4.6.2.3

Cancel the common factor of $m$ .

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Step 4.6.2.3.1

Cancel the common factor.

$\frac{m t}{m} = \frac{\ln (\frac{m}{c y} - \frac{1}{c})}{- k m}$

Step 4.6.2.3.2

Divide $t$ by $1$ .

$t = \frac{\ln (\frac{m}{c y} - \frac{1}{c})}{- k m}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Move the negative in front of the fraction.

$t = - \frac{\ln (\frac{m}{c y} - \frac{1}{c})}{k m}$