Calculus Examples

$\frac{d M}{d t} + \frac{M}{37.5} = 4$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Solve for $\frac{d M}{d t}$ .

Tap for more steps...

Step 1.1.1

Simplify each term.

Tap for more steps...

Step 1.1.1.1

Multiply by $1$ .

$\frac{d M}{d t} + \frac{1 M}{37.5} = 4$

Step 1.1.1.2

Factor $37.5$ out of $37.5$ .

$\frac{d M}{d t} + \frac{1 M}{37.5 (1)} = 4$

Step 1.1.1.3

Separate fractions.

$\frac{d M}{d t} + \frac{1}{37.5} \cdot \frac{M}{1} = 4$

Step 1.1.1.4

Divide $1$ by $37.5$ .

$\frac{d M}{d t} + 0.02\overline{6} \frac{M}{1} = 4$

Step 1.1.1.5

Divide $M$ by $1$ .

$\frac{d M}{d t} + 0.02\overline{6} M = 4$

Step 1.1.2

Subtract $0.02\overline{6} M$ from both sides of the equation.

$\frac{d M}{d t} = 4 - 0.02\overline{6} M$

Step 1.2

Multiply both sides by $\frac{1}{4 - 0.02\overline{6} M}$ .

$\frac{1}{4 - 0.02\overline{6} M} \frac{d M}{d t} = \frac{1}{4 - 0.02\overline{6} M} (4 - 0.02\overline{6} M)$

Step 1.3

Cancel the common factor of $4 - 0.02\overline{6} M$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

$\frac{1}{4 - 0.02\overline{6} M} \frac{d M}{d t} = \frac{1}{4 - 0.02\overline{6} M} (4 - 0.02\overline{6} M)$

Step 1.3.2

Rewrite the expression.

$\frac{1}{4 - 0.02\overline{6} M} \frac{d M}{d t} = 1$

Step 1.4

Rewrite the equation.

$\frac{1}{4 - 0.02\overline{6} M} d M = 1 d t$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{4 - 0.02\overline{6} M} d M = \int d t$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Let $u = 4 - 0.02\overline{6} M$ . Then $d u = - 0.02\overline{6} d M$ , so $\frac{1}{- 0.02\overline{6}} d u = d M$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.2.1.1

Let $u = 4 - 0.02\overline{6} M$ . Find $\frac{d u}{d M}$ .

Tap for more steps...

Step 2.2.1.1.1

Differentiate $4 - 0.02\overline{6} M$ .

$\frac{d}{d M} [4 - 0.02\overline{6} M]$

Step 2.2.1.1.2

Differentiate.

Tap for more steps...

Step 2.2.1.1.2.1

By the Sum Rule, the derivative of $4 - 0.02\overline{6} M$ with respect to $M$ is $\frac{d}{d M} [4] + \frac{d}{d M} [- 0.02\overline{6} M]$ .

$\frac{d}{d M} [4] + \frac{d}{d M} [- 0.02\overline{6} M]$

Step 2.2.1.1.2.2

Since $4$ is constant with respect to $M$ , the derivative of $4$ with respect to $M$ is $0$ .

$0 + \frac{d}{d M} [- 0.02\overline{6} M]$

Step 2.2.1.1.3

Evaluate $\frac{d}{d M} [- 0.02\overline{6} M]$ .

Tap for more steps...

Step 2.2.1.1.3.1

Since $- 0.02\overline{6}$ is constant with respect to $M$ , the derivative of $- 0.02\overline{6} M$ with respect to $M$ is $- 0.02\overline{6} \frac{d}{d M} [M]$ .

$0 - 0.02\overline{6} \frac{d}{d M} [M]$

Step 2.2.1.1.3.2

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

$0 - 0.02\overline{6} \cdot 1$

Step 2.2.1.1.3.3

Multiply $- 0.02\overline{6}$ by $1$ .

$0 - 0.02\overline{6}$

Step 2.2.1.1.4

Subtract $0.02\overline{6}$ from $0$ .

$- 0.02\overline{6}$

Step 2.2.1.2

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

$\int \frac{1}{u} \cdot \frac{1}{- 0.02\overline{6}} d u = \int d t$

Step 2.2.2

Simplify.

Tap for more steps...

Step 2.2.2.1

Move the negative in front of the fraction.

$\int \frac{1}{u} (- \frac{1}{0.02\overline{6}}) d u = \int d t$

Step 2.2.2.2

Multiply $\frac{1}{u}$ by $\frac{1}{0.02\overline{6}}$ .

$\int - \frac{1}{u \cdot 0.02\overline{6}} d u = \int d t$

Step 2.2.2.3

Move $0.02\overline{6}$ to the left of $u$ .

$\int - \frac{1}{0.02\overline{6} u} d u = \int d t$

Step 2.2.3

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

$- \int \frac{1}{0.02\overline{6} u} d u = \int d t$

Step 2.2.4

Since $\frac{1}{0.02\overline{6}}$ is constant with respect to $u$ , move $\frac{1}{0.02\overline{6}}$ out of the integral.

$- (\frac{1}{0.02\overline{6}} \int \frac{1}{u} d u) = \int d t$

Step 2.2.5

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

$- \frac{1}{0.02\overline{6}} (\ln (| u |) + C_{1}) = \int d t$

Step 2.2.6

Simplify.

Tap for more steps...

Step 2.2.6.1

Rewrite $- \frac{1}{0.02\overline{6}} (\ln (| u |) + C_{1})$ as $- 1 \cdot 37.5 \ln (| u |) + C_{1}$ .

$- 1 \cdot 37.5 \ln (| u |) + C_{1} = \int d t$

Step 2.2.6.2

Multiply $- 1$ by $37.5$ .

$- 37.5 \ln (| u |) + C_{1} = \int d t$

Step 2.2.7

Replace all occurrences of $u$ with $4 - 0.02\overline{6} M$ .

$- 37.5 \ln (| 4 - 0.02\overline{6} M |) + C_{1} = \int d t$

Step 2.3

Apply the constant rule.

$- 37.5 \ln (| 4 - 0.02\overline{6} M |) + C_{1} = t + C_{2}$

Step 2.4

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

$- 37.5 \ln (| 4 - 0.02\overline{6} M |) = t + C$

Step 3

Solve for $M$ .

Tap for more steps...

Step 3.1

Divide each term in $- 37.5 \ln (| 4 - 0.02\overline{6} M |) = t + C$ by $- 37.5$ and simplify.

Tap for more steps...

Step 3.1.1

Divide each term in $- 37.5 \ln (| 4 - 0.02\overline{6} M |) = t + C$ by $- 37.5$ .

$\frac{- 37.5 \ln (| 4 - 0.02\overline{6} M |)}{- 37.5} = \frac{t}{- 37.5} + \frac{C}{- 37.5}$

Step 3.1.2

Simplify the left side.

Tap for more steps...

Step 3.1.2.1

Cancel the common factor of $- 37.5$ .

Tap for more steps...

Step 3.1.2.1.1

Cancel the common factor.

$\frac{- 37.5 \ln (| 4 - 0.02\overline{6} M |)}{- 37.5} = \frac{t}{- 37.5} + \frac{C}{- 37.5}$

Step 3.1.2.1.2

Divide $\ln (| 4 - 0.02\overline{6} M |)$ by $1$ .

$\ln (| 4 - 0.02\overline{6} M |) = \frac{t}{- 37.5} + \frac{C}{- 37.5}$

Step 3.1.3

Simplify the right side.

Tap for more steps...

Step 3.1.3.1

Simplify each term.

Tap for more steps...

Step 3.1.3.1.1

Move the negative in front of the fraction.

$\ln (| 4 - 0.02\overline{6} M |) = - \frac{t}{37.5} + \frac{C}{- 37.5}$

Step 3.1.3.1.2

Multiply by $1$ .

$\ln (| 4 - 0.02\overline{6} M |) = - \frac{1 t}{37.5} + \frac{C}{- 37.5}$

Step 3.1.3.1.3

Factor $37.5$ out of $37.5$ .

$\ln (| 4 - 0.02\overline{6} M |) = - \frac{1 t}{37.5 (1)} + \frac{C}{- 37.5}$

Step 3.1.3.1.4

Separate fractions.

$\ln (| 4 - 0.02\overline{6} M |) = - (\frac{1}{37.5} \cdot \frac{t}{1}) + \frac{C}{- 37.5}$

Step 3.1.3.1.5

Divide $1$ by $37.5$ .

$\ln (| 4 - 0.02\overline{6} M |) = - (0.02\overline{6} \frac{t}{1}) + \frac{C}{- 37.5}$

Step 3.1.3.1.6

Divide $t$ by $1$ .

$\ln (| 4 - 0.02\overline{6} M |) = - (0.02\overline{6} t) + \frac{C}{- 37.5}$

Step 3.1.3.1.7

Multiply $0.02\overline{6}$ by $- 1$ .

$\ln (| 4 - 0.02\overline{6} M |) = - 0.02\overline{6} t + \frac{C}{- 37.5}$

Step 3.1.3.1.8

Move the negative in front of the fraction.

$\ln (| 4 - 0.02\overline{6} M |) = - 0.02\overline{6} t - \frac{C}{37.5}$

Step 3.1.3.1.9

Multiply by $1$ .

$\ln (| 4 - 0.02\overline{6} M |) = - 0.02\overline{6} t - \frac{1 C}{37.5}$

Step 3.1.3.1.10

Factor $37.5$ out of $37.5$ .

$\ln (| 4 - 0.02\overline{6} M |) = - 0.02\overline{6} t - \frac{1 C}{37.5 (1)}$

Step 3.1.3.1.11

Separate fractions.

$\ln (| 4 - 0.02\overline{6} M |) = - 0.02\overline{6} t - (\frac{1}{37.5} \cdot \frac{C}{1})$

Step 3.1.3.1.12

Divide $1$ by $37.5$ .

$\ln (| 4 - 0.02\overline{6} M |) = - 0.02\overline{6} t - (0.02\overline{6} \frac{C}{1})$

Step 3.1.3.1.13

Divide $C$ by $1$ .

$\ln (| 4 - 0.02\overline{6} M |) = - 0.02\overline{6} t - (0.02\overline{6} C)$

Step 3.1.3.1.14

Multiply $0.02\overline{6}$ by $- 1$ .

$\ln (| 4 - 0.02\overline{6} M |) = - 0.02\overline{6} t - 0.02\overline{6} C$

Step 3.2

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

$e^{\ln (| 4 - 0.02\overline{6} M |)} = e^{- 0.02\overline{6} t - 0.02\overline{6} C}$

Step 3.3

Rewrite $\ln (| 4 - 0.02\overline{6} M |) = - 0.02\overline{6} t - 0.02\overline{6} C$ 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 $b^{y} = x$ .

$e^{- 0.02\overline{6} t - 0.02\overline{6} C} = | 4 - 0.02\overline{6} M |$

Step 3.4

Solve for $M$ .

Tap for more steps...

Step 3.4.1

Rewrite the equation as $| 4 - 0.02\overline{6} M | = e^{- 0.02\overline{6} t - 0.02\overline{6} C}$ .

$| 4 - 0.02\overline{6} M | = e^{- 0.02\overline{6} t - 0.02\overline{6} C}$

Step 3.4.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$4 - 0.02\overline{6} M = \pm e^{- 0.02\overline{6} t - 0.02\overline{6} C}$

Step 3.4.3

Subtract $4$ from both sides of the equation.

$- 0.02\overline{6} M = \pm e^{- 0.02\overline{6} t - 0.02\overline{6} C} - 4$

Step 3.4.4

Divide each term in $- 0.02\overline{6} M = \pm e^{- 0.02\overline{6} t - 0.02\overline{6} C} - 4$ by $- 0.02\overline{6}$ and simplify.

Tap for more steps...

Step 3.4.4.1

Divide each term in $- 0.02\overline{6} M = \pm e^{- 0.02\overline{6} t - 0.02\overline{6} C} - 4$ by $- 0.02\overline{6}$ .

$\frac{- 0.02\overline{6} M}{- 0.02\overline{6}} = \frac{\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C}}{- 0.02\overline{6}} + \frac{- 4}{- 0.02\overline{6}}$

Step 3.4.4.2

Simplify the left side.

Tap for more steps...

Step 3.4.4.2.1

Cancel the common factor of $- 0.02\overline{6}$ .

Tap for more steps...

Step 3.4.4.2.1.1

Cancel the common factor.

$\frac{- 0.02\overline{6} M}{- 0.02\overline{6}} = \frac{\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C}}{- 0.02\overline{6}} + \frac{- 4}{- 0.02\overline{6}}$

Step 3.4.4.2.1.2

Divide $M$ by $1$ .

$M = \frac{\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C}}{- 0.02\overline{6}} + \frac{- 4}{- 0.02\overline{6}}$

Step 3.4.4.3

Simplify the right side.

Tap for more steps...

Step 3.4.4.3.1

Simplify each term.

Tap for more steps...

Step 3.4.4.3.1.1

Simplify $\frac{\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C}}{- 0.02\overline{6}}$ .

$M = \frac{\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C}}{0.02\overline{6}} + \frac{- 4}{- 0.02\overline{6}}$

Step 3.4.4.3.1.2

Multiply by $1$ .

$M = \frac{1 (\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C})}{0.02\overline{6}} + \frac{- 4}{- 0.02\overline{6}}$

Step 3.4.4.3.1.3

Factor $0.02\overline{6}$ out of $0.02\overline{6}$ .

$M = \frac{1 (\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C})}{0.02\overline{6} (1)} + \frac{- 4}{- 0.02\overline{6}}$

Step 3.4.4.3.1.4

Separate fractions.

$M = \frac{1}{0.02\overline{6}} \cdot \frac{\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C}}{1} + \frac{- 4}{- 0.02\overline{6}}$

Step 3.4.4.3.1.5

Divide $1$ by $0.02\overline{6}$ .

$M = 37.5 \frac{\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C}}{1} + \frac{- 4}{- 0.02\overline{6}}$

Step 3.4.4.3.1.6

Divide $\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C}$ by $1$ .

$M = 37.5 (\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C}) + \frac{- 4}{- 0.02\overline{6}}$

Step 3.4.4.3.1.7

Divide $- 4$ by $- 0.02\overline{6}$ .

$M = 37.5 (\pm e^{- 0.02\overline{6} t - 0.02\overline{6} C}) + 150$

Step 4

Group the constant terms together.

Tap for more steps...

Step 4.1

Simplify the constant of integration.

$M = 37.5 (\pm e^{- 0.02\overline{6} t + C}) + 150$

Step 4.2

Rewrite $e^{- 0.02\overline{6} t + C}$ as $e^{- 0.02\overline{6} t} e^{C}$ .

$M = 37.5 (\pm e^{- 0.02\overline{6} t} e^{C}) + 150$

Step 4.3

Reorder $e^{- 0.02\overline{6} t}$ and $e^{C}$ .

$M = 37.5 (\pm e^{C} e^{- 0.02\overline{6} t}) + 150$

Step 4.4

Combine constants with the plus or minus.

$M = 37.5 (C e^{- 0.02\overline{6} t}) + 150$