Calculus Examples

$\frac{d x}{d t} = 20 - \frac{8 x}{200}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{20 - \frac{8 x}{200}}$ .

$\frac{1}{20 - \frac{8 x}{200}} \frac{d x}{d t} = \frac{1}{20 - \frac{8 x}{200}} (20 - \frac{8 x}{200})$

Step 1.2

Cancel the common factor of $20 - \frac{8 x}{200}$ .

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Step 1.2.1

Cancel the common factor.

$\frac{1}{20 - \frac{8 x}{200}} \frac{d x}{d t} = \frac{1}{20 - \frac{8 x}{200}} (20 - \frac{8 x}{200})$

Step 1.2.2

Rewrite the expression.

$\frac{1}{20 - \frac{8 x}{200}} \frac{d x}{d t} = 1$

Step 1.3

Rewrite the equation.

$\frac{1}{20 - \frac{8 x}{200}} d x = 1 d t$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{20 - \frac{8 x}{200}} d x = \int d t$

Step 2.2

Integrate the left side.

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Step 2.2.1

Cancel the common factor of $8$ and $200$ .

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Step 2.2.1.1

Factor $8$ out of $8 x$ .

$\int \frac{1}{20 - \frac{8 (x)}{200}} d x = \int d t$

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Factor $8$ out of $200$ .

$\int \frac{1}{20 - \frac{8 x}{8 \cdot 25}} d x = \int d t$

Step 2.2.1.2.2

Cancel the common factor.

$\int \frac{1}{20 - \frac{8 x}{8 \cdot 25}} d x = \int d t$

Step 2.2.1.2.3

Rewrite the expression.

$\int \frac{1}{20 - \frac{x}{25}} d x = \int d t$

Step 2.2.2

Let $u = 20 - \frac{x}{25}$ . Then $d u = - \frac{1}{25} d x$ , so $- 25 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.2.2.1

Let $u = 20 - \frac{x}{25}$ . Find $\frac{d u}{d x}$ .

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Step 2.2.2.1.1

Differentiate $20 - \frac{x}{25}$ .

$\frac{d}{d x} [20 - \frac{x}{25}]$

Step 2.2.2.1.2

Differentiate.

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Step 2.2.2.1.2.1

By the Sum Rule, the derivative of $20 - \frac{x}{25}$ with respect to $x$ is $\frac{d}{d x} [20] + \frac{d}{d x} [- \frac{x}{25}]$ .

$\frac{d}{d x} [20] + \frac{d}{d x} [- \frac{x}{25}]$

Step 2.2.2.1.2.2

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{x}{25}]$

Step 2.2.2.1.3

Evaluate $\frac{d}{d x} [- \frac{x}{25}]$ .

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Step 2.2.2.1.3.1

Since $- \frac{1}{25}$ is constant with respect to $x$ , the derivative of $- \frac{x}{25}$ with respect to $x$ is $- \frac{1}{25} \frac{d}{d x} [x]$ .

$0 - \frac{1}{25} \frac{d}{d x} [x]$

Step 2.2.2.1.3.2

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

$0 - \frac{1}{25} \cdot 1$

Step 2.2.2.1.3.3

Multiply $- 1$ by $1$ .

$0 - \frac{1}{25}$

Step 2.2.2.1.4

Subtract $\frac{1}{25}$ from $0$ .

$- \frac{1}{25}$

Step 2.2.2.2

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

$\int \frac{- 1}{u} \cdot \frac{- 1}{- \frac{1}{25}} d u = \int d t$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Move the negative in front of the fraction.

$\int - \frac{1}{u} \cdot \frac{- 1}{- \frac{1}{25}} d u = \int d t$

Step 2.2.3.2

Dividing two negative values results in a positive value.

$\int - \frac{1}{u} \cdot \frac{1}{\frac{1}{25}} d u = \int d t$

Step 2.2.3.3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{25}$ .

$\int - \frac{1}{u} (1 \cdot 25) d u = \int d t$

Step 2.2.3.4

Multiply $25$ by $1$ .

$\int - \frac{1}{u} \cdot 25 d u = \int d t$

Step 2.2.3.5

Multiply $25$ by $- 1$ .

$\int - 25 \frac{1}{u} d u = \int d t$

Step 2.2.3.6

Combine $- 25$ and $\frac{1}{u}$ .

$\int \frac{- 25}{u} d u = \int d t$

Step 2.2.3.7

Move the negative in front of the fraction.

$\int - \frac{25}{u} d u = \int d t$

Step 2.2.4

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

$- \int \frac{25}{u} d u = \int d t$

Step 2.2.5

Since $25$ is constant with respect to $u$ , move $25$ out of the integral.

$- (25 \int \frac{1}{u} d u) = \int d t$

Step 2.2.6

Multiply $25$ by $- 1$ .

$- 25 \int \frac{1}{u} d u = \int d t$

Step 2.2.7

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

$- 25 (\ln (| u |) + C_{1}) = \int d t$

Step 2.2.8

Simplify.

$- 25 \ln (| u |) + C_{1} = \int d t$

Step 2.2.9

Replace all occurrences of $u$ with $20 - \frac{x}{25}$ .

$- 25 \ln (| 20 - \frac{x}{25} |) + C_{1} = \int d t$

Step 2.2.10

Reorder terms.

$- 25 \ln (| 20 - \frac{1}{25} x |) + C_{1} = \int d t$

Step 2.3

Apply the constant rule.

$- 25 \ln (| 20 - \frac{1}{25} x |) + C_{1} = t + C_{2}$

Step 2.4

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

$- 25 \ln (| 20 - \frac{1}{25} x |) = t + C$

Step 3

Solve for $x$ .

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Step 3.1

Divide each term in $- 25 \ln (| 20 - \frac{1}{25} x |) = t + C$ by $- 25$ and simplify.

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Step 3.1.1

Divide each term in $- 25 \ln (| 20 - \frac{1}{25} x |) = t + C$ by $- 25$ .

$\frac{- 25 \ln (| 20 - \frac{1}{25} x |)}{- 25} = \frac{t}{- 25} + \frac{C}{- 25}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $- 25$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{- 25 \ln (| 20 - \frac{1}{25} x |)}{- 25} = \frac{t}{- 25} + \frac{C}{- 25}$

Step 3.1.2.1.2

Divide $\ln (| 20 - \frac{1}{25} x |)$ by $1$ .

$\ln (| 20 - \frac{1}{25} x |) = \frac{t}{- 25} + \frac{C}{- 25}$

Step 3.1.2.2

Combine $x$ and $\frac{1}{25}$ .

$\ln (| 20 - \frac{x}{25} |) = \frac{t}{- 25} + \frac{C}{- 25}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

Move the negative in front of the fraction.

$\ln (| 20 - \frac{x}{25} |) = - \frac{t}{25} + \frac{C}{- 25}$

Step 3.1.3.1.2

Move the negative in front of the fraction.

$\ln (| 20 - \frac{x}{25} |) = - \frac{t}{25} - \frac{C}{25}$

Step 3.2

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

$e^{\ln (| 20 - \frac{x}{25} |)} = e^{- \frac{t}{25} - \frac{C}{25}}$

Step 3.3

Rewrite $\ln (| 20 - \frac{x}{25} |) = - \frac{t}{25} - \frac{C}{25}$ 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 $b^{y} = x$ .

$e^{- \frac{t}{25} - \frac{C}{25}} = | 20 - \frac{x}{25} |$

Step 3.4

Solve for $x$ .

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Step 3.4.1

Rewrite the equation as $| 20 - \frac{x}{25} | = e^{- \frac{t}{25} - \frac{C}{25}}$ .

$| 20 - \frac{x}{25} | = e^{- \frac{t}{25} - \frac{C}{25}}$

Step 3.4.2

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

$20 - \frac{x}{25} = \pm e^{- \frac{t}{25} - \frac{C}{25}}$

Step 3.4.3

Subtract $20$ from both sides of the equation.

$- \frac{x}{25} = \pm e^{- \frac{t}{25} - \frac{C}{25}} - 20$

Step 3.4.4

Multiply both sides of the equation by $- 25$ .

$- 25 (- \frac{x}{25}) = - 25 (\pm e^{- \frac{t}{25} - \frac{C}{25}} - 20)$

Step 3.4.5

Simplify both sides of the equation.

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Step 3.4.5.1

Simplify the left side.

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Step 3.4.5.1.1

Simplify $- 25 (- \frac{x}{25})$ .

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Step 3.4.5.1.1.1

Cancel the common factor of $25$ .

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Step 3.4.5.1.1.1.1

Move the leading negative in $- \frac{x}{25}$ into the numerator.

$- 25 \frac{- x}{25} = - 25 (\pm e^{- \frac{t}{25} - \frac{C}{25}} - 20)$

Step 3.4.5.1.1.1.2

Factor $25$ out of $- 25$ .

$25 (- 1) \frac{- x}{25} = - 25 (\pm e^{- \frac{t}{25} - \frac{C}{25}} - 20)$

Step 3.4.5.1.1.1.3

Cancel the common factor.

$25 \cdot - 1 \frac{- x}{25} = - 25 (\pm e^{- \frac{t}{25} - \frac{C}{25}} - 20)$

Step 3.4.5.1.1.1.4

Rewrite the expression.

$- - x = - 25 (\pm e^{- \frac{t}{25} - \frac{C}{25}} - 20)$

Step 3.4.5.1.1.2

Multiply.

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Step 3.4.5.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x = - 25 (\pm e^{- \frac{t}{25} - \frac{C}{25}} - 20)$

Step 3.4.5.1.1.2.2

Multiply $x$ by $1$ .

$x = - 25 (\pm e^{- \frac{t}{25} - \frac{C}{25}} - 20)$

Step 3.4.5.2

Simplify the right side.

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Step 3.4.5.2.1

Simplify $- 25 (\pm e^{- \frac{t}{25} - \frac{C}{25}} - 20)$ .

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Step 3.4.5.2.1.1

Apply the distributive property.

$x = - 25 (\pm e^{- \frac{t}{25} - \frac{C}{25}}) - 25 \cdot - 20$

Step 3.4.5.2.1.2

Multiply $- 25$ by $- 20$ .

$x = - 25 (\pm e^{- \frac{t}{25} - \frac{C}{25}}) + 500$

Step 3.4.6

Reorder $- \frac{t}{25}$ and $- \frac{C}{25}$ .

$x = - 25 (\pm e^{- \frac{C}{25} - \frac{t}{25}}) + 500$

Step 4

Group the constant terms together.

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Step 4.1

Simplify the constant of integration.

$x = - 25 (\pm e^{C - \frac{t}{25}}) + 500$

Step 4.2

Reorder terms.

$x = - 25 (\pm e^{- \frac{t}{25} + C}) + 500$

Step 4.3

Rewrite $e^{- \frac{t}{25} + C}$ as $e^{- \frac{t}{25}} e^{C}$ .

$x = - 25 (\pm e^{- \frac{t}{25}} e^{C}) + 500$

Step 4.4

Reorder $e^{- \frac{t}{25}}$ and $e^{C}$ .

$x = - 25 (\pm e^{C} e^{- \frac{t}{25}}) + 500$

Step 4.5

Combine constants with the plus or minus.

$x = - 25 (C e^{- \frac{t}{25}}) + 500$