Calculus Examples

$\frac{d A}{d t} + \frac{A}{100} = 6$

Step 1

Separate the variables.

Tap for more steps...

Step 1.1

Subtract $\frac{A}{100}$ from both sides of the equation.

$\frac{d A}{d t} = 6 - \frac{A}{100}$

Step 1.2

Multiply both sides by $\frac{1}{6 - \frac{A}{100}}$ .

$\frac{1}{6 - \frac{A}{100}} \frac{d A}{d t} = \frac{1}{6 - \frac{A}{100}} (6 - \frac{A}{100})$

Step 1.3

Cancel the common factor of $6 - \frac{A}{100}$ .

Tap for more steps...

Step 1.3.1

Cancel the common factor.

$\frac{1}{6 - \frac{A}{100}} \frac{d A}{d t} = \frac{1}{6 - \frac{A}{100}} (6 - \frac{A}{100})$

Step 1.3.2

Rewrite the expression.

$\frac{1}{6 - \frac{A}{100}} \frac{d A}{d t} = 1$

Step 1.4

Rewrite the equation.

$\frac{1}{6 - \frac{A}{100}} d A = 1 d t$

Step 2

Integrate both sides.

Tap for more steps...

Step 2.1

Set up an integral on each side.

$\int \frac{1}{6 - \frac{A}{100}} d A = \int d t$

Step 2.2

Integrate the left side.

Tap for more steps...

Step 2.2.1

Let $u = 6 - \frac{A}{100}$ . Then $d u = - \frac{1}{100} d A$ , so $- 100 d u = d A$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 2.2.1.1

Let $u = 6 - \frac{A}{100}$ . Find $\frac{d u}{d A}$ .

Tap for more steps...

Step 2.2.1.1.1

Differentiate $6 - \frac{A}{100}$ .

$\frac{d}{d A} [6 - \frac{A}{100}]$

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 $6 - \frac{A}{100}$ with respect to $A$ is $\frac{d}{d A} [6] + \frac{d}{d A} [- \frac{A}{100}]$ .

$\frac{d}{d A} [6] + \frac{d}{d A} [- \frac{A}{100}]$

Step 2.2.1.1.2.2

Since $6$ is constant with respect to $A$ , the derivative of $6$ with respect to $A$ is $0$ .

$0 + \frac{d}{d A} [- \frac{A}{100}]$

Step 2.2.1.1.3

Evaluate $\frac{d}{d A} [- \frac{A}{100}]$ .

Tap for more steps...

Step 2.2.1.1.3.1

Since $- \frac{1}{100}$ is constant with respect to $A$ , the derivative of $- \frac{A}{100}$ with respect to $A$ is $- \frac{1}{100} \frac{d}{d A} [A]$ .

$0 - \frac{1}{100} \frac{d}{d A} [A]$

Step 2.2.1.1.3.2

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

$0 - \frac{1}{100} \cdot 1$

Step 2.2.1.1.3.3

Multiply $- 1$ by $1$ .

$0 - \frac{1}{100}$

Step 2.2.1.1.4

Subtract $\frac{1}{100}$ from $0$ .

$- \frac{1}{100}$

Step 2.2.1.2

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

$\int \frac{- 1}{u} \cdot \frac{- 1}{- \frac{1}{100}} d u = \int d t$

Step 2.2.2

Simplify.

Tap for more steps...

Step 2.2.2.1

Move the negative in front of the fraction.

$\int - \frac{1}{u} \cdot \frac{- 1}{- \frac{1}{100}} d u = \int d t$

Step 2.2.2.2

Dividing two negative values results in a positive value.

$\int - \frac{1}{u} \cdot \frac{1}{\frac{1}{100}} d u = \int d t$

Step 2.2.2.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{100}$ .

$\int - \frac{1}{u} (1 \cdot 100) d u = \int d t$

Step 2.2.2.4

Multiply $100$ by $1$ .

$\int - \frac{1}{u} \cdot 100 d u = \int d t$

Step 2.2.2.5

Multiply $100$ by $- 1$ .

$\int - 100 \frac{1}{u} d u = \int d t$

Step 2.2.2.6

Combine $- 100$ and $\frac{1}{u}$ .

$\int \frac{- 100}{u} d u = \int d t$

Step 2.2.2.7

Move the negative in front of the fraction.

$\int - \frac{100}{u} d u = \int d t$

Step 2.2.3

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

$- \int \frac{100}{u} d u = \int d t$

Step 2.2.4

Since $100$ is constant with respect to $u$ , move $100$ out of the integral.

$- (100 \int \frac{1}{u} d u) = \int d t$

Step 2.2.5

Multiply $100$ by $- 1$ .

$- 100 \int \frac{1}{u} d u = \int d t$

Step 2.2.6

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

$- 100 (\ln (| u |) + C_{1}) = \int d t$

Step 2.2.7

Simplify.

$- 100 \ln (| u |) + C_{1} = \int d t$

Step 2.2.8

Replace all occurrences of $u$ with $6 - \frac{A}{100}$ .

$- 100 \ln (| 6 - \frac{A}{100} |) + C_{1} = \int d t$

Step 2.2.9

Reorder terms.

$- 100 \ln (| 6 - \frac{1}{100} A |) + C_{1} = \int d t$

Step 2.3

Apply the constant rule.

$- 100 \ln (| 6 - \frac{1}{100} A |) + C_{1} = t + C_{2}$

Step 2.4

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

$- 100 \ln (| 6 - \frac{1}{100} A |) = t + C$

Step 3

Solve for $A$ .

Tap for more steps...

Step 3.1

Divide each term in $- 100 \ln (| 6 - \frac{1}{100} A |) = t + C$ by $- 100$ and simplify.

Tap for more steps...

Step 3.1.1

Divide each term in $- 100 \ln (| 6 - \frac{1}{100} A |) = t + C$ by $- 100$ .

$\frac{- 100 \ln (| 6 - \frac{1}{100} A |)}{- 100} = \frac{t}{- 100} + \frac{C}{- 100}$

Step 3.1.2

Simplify the left side.

Tap for more steps...

Step 3.1.2.1

Cancel the common factor of $- 100$ .

Tap for more steps...

Step 3.1.2.1.1

Cancel the common factor.

$\frac{- 100 \ln (| 6 - \frac{1}{100} A |)}{- 100} = \frac{t}{- 100} + \frac{C}{- 100}$

Step 3.1.2.1.2

Divide $\ln (| 6 - \frac{1}{100} A |)$ by $1$ .

$\ln (| 6 - \frac{1}{100} A |) = \frac{t}{- 100} + \frac{C}{- 100}$

Step 3.1.2.2

Combine $A$ and $\frac{1}{100}$ .

$\ln (| 6 - \frac{A}{100} |) = \frac{t}{- 100} + \frac{C}{- 100}$

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.

$\ln (| 6 - \frac{A}{100} |) = - \frac{t}{100} + \frac{C}{- 100}$

Step 3.1.3.1.2

Move the negative in front of the fraction.

$\ln (| 6 - \frac{A}{100} |) = - \frac{t}{100} - \frac{C}{100}$

Step 3.2

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

$e^{\ln (| 6 - \frac{A}{100} |)} = e^{- \frac{t}{100} - \frac{C}{100}}$

Step 3.3

Rewrite $\ln (| 6 - \frac{A}{100} |) = - \frac{t}{100} - \frac{C}{100}$ 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 $b^{y} = x$ .

$e^{- \frac{t}{100} - \frac{C}{100}} = | 6 - \frac{A}{100} |$

Step 3.4

Solve for $A$ .

Tap for more steps...

Step 3.4.1

Rewrite the equation as $| 6 - \frac{A}{100} | = e^{- \frac{t}{100} - \frac{C}{100}}$ .

$| 6 - \frac{A}{100} | = e^{- \frac{t}{100} - \frac{C}{100}}$

Step 3.4.2

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

$6 - \frac{A}{100} = \pm e^{- \frac{t}{100} - \frac{C}{100}}$

Step 3.4.3

Subtract $6$ from both sides of the equation.

$- \frac{A}{100} = \pm e^{- \frac{t}{100} - \frac{C}{100}} - 6$

Step 3.4.4

Multiply both sides of the equation by $- 100$ .

$- 100 (- \frac{A}{100}) = - 100 (\pm e^{- \frac{t}{100} - \frac{C}{100}} - 6)$

Step 3.4.5

Simplify both sides of the equation.

Tap for more steps...

Step 3.4.5.1

Simplify the left side.

Tap for more steps...

Step 3.4.5.1.1

Simplify $- 100 (- \frac{A}{100})$ .

Tap for more steps...

Step 3.4.5.1.1.1

Cancel the common factor of $100$ .

Tap for more steps...

Step 3.4.5.1.1.1.1

Move the leading negative in $- \frac{A}{100}$ into the numerator.

$- 100 \frac{- A}{100} = - 100 (\pm e^{- \frac{t}{100} - \frac{C}{100}} - 6)$

Step 3.4.5.1.1.1.2

Factor $100$ out of $- 100$ .

$100 (- 1) \frac{- A}{100} = - 100 (\pm e^{- \frac{t}{100} - \frac{C}{100}} - 6)$

Step 3.4.5.1.1.1.3

Cancel the common factor.

$100 \cdot - 1 \frac{- A}{100} = - 100 (\pm e^{- \frac{t}{100} - \frac{C}{100}} - 6)$

Step 3.4.5.1.1.1.4

Rewrite the expression.

$- - A = - 100 (\pm e^{- \frac{t}{100} - \frac{C}{100}} - 6)$

Step 3.4.5.1.1.2

Multiply.

Tap for more steps...

Step 3.4.5.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 A = - 100 (\pm e^{- \frac{t}{100} - \frac{C}{100}} - 6)$

Step 3.4.5.1.1.2.2

Multiply $A$ by $1$ .

$A = - 100 (\pm e^{- \frac{t}{100} - \frac{C}{100}} - 6)$

Step 3.4.5.2

Simplify the right side.

Tap for more steps...

Step 3.4.5.2.1

Simplify $- 100 (\pm e^{- \frac{t}{100} - \frac{C}{100}} - 6)$ .

Tap for more steps...

Step 3.4.5.2.1.1

Apply the distributive property.

$A = - 100 (\pm e^{- \frac{t}{100} - \frac{C}{100}}) - 100 \cdot - 6$

Step 3.4.5.2.1.2

Multiply $- 100$ by $- 6$ .

$A = - 100 (\pm e^{- \frac{t}{100} - \frac{C}{100}}) + 600$

Step 3.4.6

Reorder $- \frac{t}{100}$ and $- \frac{C}{100}$ .

$A = - 100 (\pm e^{- \frac{C}{100} - \frac{t}{100}}) + 600$

Step 4

Group the constant terms together.

Tap for more steps...

Step 4.1

Simplify the constant of integration.

$A = - 100 (\pm e^{C - \frac{t}{100}}) + 600$

Step 4.2

Reorder terms.

$A = - 100 (\pm e^{- \frac{t}{100} + C}) + 600$

Step 4.3

Rewrite $e^{- \frac{t}{100} + C}$ as $e^{- \frac{t}{100}} e^{C}$ .

$A = - 100 (\pm e^{- \frac{t}{100}} e^{C}) + 600$

Step 4.4

Reorder $e^{- \frac{t}{100}}$ and $e^{C}$ .

$A = - 100 (\pm e^{C} e^{- \frac{t}{100}}) + 600$

Step 4.5

Combine constants with the plus or minus.

$A = - 100 (C e^{- \frac{t}{100}}) + 600$