Calculus Examples

$\frac{d x}{d t} = e^{x - t}$

Step 1

Let $v = e^{x - t}$ . Substitute $v$ for all occurrences of $e^{x - t}$ .

$\frac{d x}{d t} = v$

Step 2

Find $\frac{d v}{d t}$ by differentiating $e^{x - t}$ .

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Step 2.1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = e^{t}$ and $g (t) = x - t$ .

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Step 2.1.1

To apply the Chain Rule, set $u_{1}$ as $x - t$ .

$\frac{d v}{d t} = \frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d t} [x - t]$

Step 2.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$\frac{d v}{d t} = e^{u_{1}} \frac{d}{d t} [x - t]$

Step 2.1.3

Replace all occurrences of $u_{1}$ with $x - t$ .

$\frac{d v}{d t} = e^{x - t} \frac{d}{d t} [x - t]$

Step 2.2

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

$\frac{d v}{d t} = e^{x - t} (\frac{d}{d t} [x] + \frac{d}{d t} [- t])$

Step 2.3

Rewrite $\frac{d}{d t} [x]$ as $x'$ .

$\frac{d v}{d t} = e^{x - t} (x' + \frac{d}{d t} [- t])$

Step 2.4

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

$\frac{d v}{d t} = e^{x - t} (x' - \frac{d}{d t} [t])$

Step 2.5

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

$\frac{d v}{d t} = e^{x - t} (x' - 1 \cdot 1)$

Step 2.6

Multiply $- 1$ by $1$ .

$\frac{d v}{d t} = e^{x - t} (x' - 1)$

Step 3

Substitute $v$ for $e^{x - t}$ .

$\frac{d v}{d t} = v (x' - 1)$

Step 4

Substitute the derivative back in to the differential equation.

$\frac{\frac{d v}{d t}}{v} + 1 = v$

Step 5

Separate the variables.

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Step 5.1

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

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Step 5.1.1

Subtract $1$ from both sides of the equation.

$\frac{\frac{d v}{d t}}{v} = v - 1$

Step 5.1.2

Multiply both sides by $v$ .

$\frac{\frac{d v}{d t}}{v} v = (v - 1) v$

Step 5.1.3

Simplify.

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Step 5.1.3.1

Simplify the left side.

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Step 5.1.3.1.1

Cancel the common factor of $v$ .

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Step 5.1.3.1.1.1

Cancel the common factor.

$\frac{\frac{d v}{d t}}{v} v = (v - 1) v$

Step 5.1.3.1.1.2

Rewrite the expression.

$\frac{d v}{d t} = (v - 1) v$

Step 5.1.3.2

Simplify the right side.

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Step 5.1.3.2.1

Simplify $(v - 1) v$ .

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Step 5.1.3.2.1.1

Apply the distributive property.

$\frac{d v}{d t} = v \cdot v - 1 v$

Step 5.1.3.2.1.2

Simplify the expression.

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Step 5.1.3.2.1.2.1

Multiply $v$ by $v$ .

$\frac{d v}{d t} = v^{2} - 1 v$

Step 5.1.3.2.1.2.2

Rewrite $- 1 v$ as $- v$ .

$\frac{d v}{d t} = v^{2} - v$

Step 5.2

Multiply both sides by $\frac{1}{v^{2} - v}$ .

$\frac{1}{v^{2} - v} \frac{d v}{d t} = \frac{1}{v^{2} - v} (v^{2} - v)$

Step 5.3

Cancel the common factor of $v^{2} - v$ .

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Step 5.3.1

Cancel the common factor.

$\frac{1}{v^{2} - v} \frac{d v}{d t} = \frac{1}{v^{2} - v} (v^{2} - v)$

Step 5.3.2

Rewrite the expression.

$\frac{1}{v^{2} - v} \frac{d v}{d t} = 1$

Step 5.4

Rewrite the equation.

$\frac{1}{v^{2} - v} d v = 1 d t$

Step 6

Integrate both sides.

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Step 6.1

Set up an integral on each side.

$\int \frac{1}{v^{2} - v} d v = \int d t$

Step 6.2

Integrate the left side.

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Step 6.2.1

Write the fraction using partial fraction decomposition.

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Step 6.2.1.1

Decompose the fraction and multiply through by the common denominator.

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Step 6.2.1.1.1

Factor $v$ out of $v^{2} - v$ .

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Step 6.2.1.1.1.1

Factor $v$ out of $v^{2}$ .

$\frac{1}{v \cdot v - v}$

Step 6.2.1.1.1.2

Factor $v$ out of $- v$ .

$\frac{1}{v \cdot v + v \cdot - 1}$

Step 6.2.1.1.1.3

Factor $v$ out of $v \cdot v + v \cdot - 1$ .

$\frac{1}{v (v - 1)}$

Step 6.2.1.1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $B$ .

$\frac{A}{v} + \frac{B}{v - 1}$

Step 6.2.1.1.3

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is $v (v - 1)$ .

$\frac{1 (v (v - 1))}{v (v - 1)} = \frac{A (v (v - 1))}{v} + \frac{B (v (v - 1))}{v - 1}$

Step 6.2.1.1.4

Cancel the common factor of $v$ .

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Step 6.2.1.1.4.1

Cancel the common factor.

$\frac{1 (v (v - 1))}{v (v - 1)} = \frac{A (v (v - 1))}{v} + \frac{B (v (v - 1))}{v - 1}$

Step 6.2.1.1.4.2

Rewrite the expression.

$\frac{1 (v - 1)}{v - 1} = \frac{A (v (v - 1))}{v} + \frac{B (v (v - 1))}{v - 1}$

Step 6.2.1.1.5

Cancel the common factor of $v - 1$ .

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Step 6.2.1.1.5.1

Cancel the common factor.

$\frac{1 (v - 1)}{v - 1} = \frac{A (v (v - 1))}{v} + \frac{B (v (v - 1))}{v - 1}$

Step 6.2.1.1.5.2

Rewrite the expression.

$1 = \frac{A (v (v - 1))}{v} + \frac{B (v (v - 1))}{v - 1}$

Step 6.2.1.1.6

Simplify each term.

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Step 6.2.1.1.6.1

Cancel the common factor of $v$ .

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Step 6.2.1.1.6.1.1

Cancel the common factor.

$1 = \frac{A (v (v - 1))}{v} + \frac{B (v (v - 1))}{v - 1}$

Step 6.2.1.1.6.1.2

Divide $A (v - 1)$ by $1$ .

$1 = A (v - 1) + \frac{B (v (v - 1))}{v - 1}$

Step 6.2.1.1.6.2

Apply the distributive property.

$1 = A (v) + A \cdot - 1 + \frac{B (v (v - 1))}{v - 1}$

Step 6.2.1.1.6.3

Move $- 1$ to the left of $A$ .

$1 = A (v) - 1 \cdot A + \frac{B (v (v - 1))}{v - 1}$

Step 6.2.1.1.6.4

Rewrite $- 1 A$ as $- A$ .

$1 = A (v) - A + \frac{B (v (v - 1))}{v - 1}$

Step 6.2.1.1.6.5

Cancel the common factor of $v - 1$ .

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Step 6.2.1.1.6.5.1

Cancel the common factor.

$1 = A (v) - A + \frac{B (v (v - 1))}{v - 1}$

Step 6.2.1.1.6.5.2

Divide $B v$ by $1$ .

$1 = A v - A + B v$

Step 6.2.1.1.7

Move $- A$ .

$1 = A v + B v - A$

Step 6.2.1.2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 6.2.1.2.1

Create an equation for the partial fraction variables by equating the coefficients of $v$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = A + B$

Step 6.2.1.2.2

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $v$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$1 = - 1 A$

Step 6.2.1.2.3

Set up the system of equations to find the coefficients of the partial fractions.

$0 = A + B$

$1 = - 1 A$

$0 = A + B$

$1 = - 1 A$

Step 6.2.1.3

Solve the system of equations.

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Step 6.2.1.3.1

Solve for $A$ in $1 = - 1 A$ .

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Step 6.2.1.3.1.1

Rewrite the equation as $- 1 A = 1$ .

$- 1 A = 1$

$0 = A + B$

Step 6.2.1.3.1.2

Divide each term in $- 1 A = 1$ by $- 1$ and simplify.

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Step 6.2.1.3.1.2.1

Divide each term in $- 1 A = 1$ by $- 1$ .

$\frac{- 1 A}{- 1} = \frac{1}{- 1}$

$0 = A + B$

Step 6.2.1.3.1.2.2

Simplify the left side.

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Step 6.2.1.3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{A}{1} = \frac{1}{- 1}$

$0 = A + B$

Step 6.2.1.3.1.2.2.2

Divide $A$ by $1$ .

$A = \frac{1}{- 1}$

$0 = A + B$

$A = \frac{1}{- 1}$

$0 = A + B$

Step 6.2.1.3.1.2.3

Simplify the right side.

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Step 6.2.1.3.1.2.3.1

Divide $1$ by $- 1$ .

$A = - 1$

$0 = A + B$

$A = - 1$

$0 = A + B$

$A = - 1$

$0 = A + B$

$A = - 1$

$0 = A + B$

Step 6.2.1.3.2

Replace all occurrences of $A$ with $- 1$ in each equation.

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Step 6.2.1.3.2.1

Replace all occurrences of $A$ in $0 = A + B$ with $- 1$ .

$0 = (- 1) + B$

$A = - 1$

Step 6.2.1.3.2.2

Simplify the right side.

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Step 6.2.1.3.2.2.1

Remove parentheses.

$0 = - 1 + B$

$A = - 1$

$0 = - 1 + B$

$A = - 1$

$0 = - 1 + B$

$A = - 1$

Step 6.2.1.3.3

Solve for $B$ in $0 = - 1 + B$ .

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Step 6.2.1.3.3.1

Rewrite the equation as $- 1 + B = 0$ .

$- 1 + B = 0$

$A = - 1$

Step 6.2.1.3.3.2

Add $1$ to both sides of the equation.

$B = 1$

$A = - 1$

$B = 1$

$A = - 1$

Step 6.2.1.3.4

Solve the system of equations.

$B = 1 A = - 1$

Step 6.2.1.3.5

List all of the solutions.

$B = 1, A = - 1$

Step 6.2.1.4

Replace each of the partial fraction coefficients in $\frac{A}{v} + \frac{B}{v - 1}$ with the values found for $A$ and $B$ .

$- \frac{1}{v} + \frac{1}{v - 1}$

Step 6.2.1.5

Move the negative in front of the fraction.

$\int - \frac{1}{v} + \frac{1}{v - 1} d v = \int d t$

Step 6.2.2

Split the single integral into multiple integrals.

$\int - \frac{1}{v} d v + \int \frac{1}{v - 1} d v = \int d t$

Step 6.2.3

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

$- \int \frac{1}{v} d v + \int \frac{1}{v - 1} d v = \int d t$

Step 6.2.4

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

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

Step 6.2.5

Let $u_{2} = v - 1$ . Then $d u_{2} = d v$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.2.5.1

Let $u_{2} = v - 1$ . Find $\frac{d u_{2}}{d v}$ .

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Step 6.2.5.1.1

Differentiate $v - 1$ .

$\frac{d}{d v} [v - 1]$

Step 6.2.5.1.2

By the Sum Rule, the derivative of $v - 1$ with respect to $v$ is $\frac{d}{d v} [v] + \frac{d}{d v} [- 1]$ .

$\frac{d}{d v} [v] + \frac{d}{d v} [- 1]$

Step 6.2.5.1.3

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

$1 + \frac{d}{d v} [- 1]$

Step 6.2.5.1.4

Since $- 1$ is constant with respect to $v$ , the derivative of $- 1$ with respect to $v$ is $0$ .

$1 + 0$

Step 6.2.5.1.5

Add $1$ and $0$ .

$1$

Step 6.2.5.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$- (\ln (| v |) + C_{1}) + \int \frac{1}{u_{2}} d u_{2} = \int d t$

Step 6.2.6

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$- (\ln (| v |) + C_{1}) + \ln (| u_{2} |) + C_{2} = \int d t$

Step 6.2.7

Simplify.

$- \ln (| v |) + \ln (| u_{2} |) + C_{3} = \int d t$

Step 6.2.8

Reorder terms.

$\ln (| u_{2} |) - \ln (| v |) + C_{3} = \int d t$

Step 6.3

Apply the constant rule.

$\ln (| u_{2} |) - \ln (| v |) + C_{3} = t + C_{4}$

Step 6.4

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

$\ln (| u_{2} |) - \ln (| v |) = t + C$

Step 7

Solve for $v$ .

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Step 7.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{| u_{2} |}{| v |}) = t + C$

Step 7.2

Reorder $t$ and $C$ .

$\ln (\frac{| u_{2} |}{| v |}) = C + t$

Step 7.3

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

$e^{\ln (\frac{| u_{2} |}{| v |})} = e^{C + t}$

Step 7.4

Rewrite $\ln (\frac{| u_{2} |}{| v |}) = C + t$ 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 $b^{y} = x$ .

$e^{C + t} = \frac{| u_{2} |}{| v |}$

Step 7.5

Solve for $v$ .

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Step 7.5.1

Rewrite the equation as $\frac{| u_{2} |}{| v |} = e^{C + t}$ .

$\frac{| u_{2} |}{| v |} = e^{C + t}$

Step 7.5.2

Multiply both sides by $| v |$ .

$\frac{| u_{2} |}{| v |} | v | = e^{C + t} | v |$

Step 7.5.3

Simplify the left side.

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Step 7.5.3.1

Cancel the common factor of $| v |$ .

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Step 7.5.3.1.1

Cancel the common factor.

$\frac{| u_{2} |}{| v |} | v | = e^{C + t} | v |$

Step 7.5.3.1.2

Rewrite the expression.

$| u_{2} | = e^{C + t} | v |$

Step 7.5.4

Solve for $v$ .

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Step 7.5.4.1

Rewrite the equation as $e^{C + t} | v | = | u_{2} |$ .

$e^{C + t} | v | = | u_{2} |$

Step 7.5.4.2

Divide each term in $e^{C + t} | v | = | u_{2} |$ by $e^{C + t}$ and simplify.

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Step 7.5.4.2.1

Divide each term in $e^{C + t} | v | = | u_{2} |$ by $e^{C + t}$ .

$\frac{e^{C + t} | v |}{e^{C + t}} = \frac{| u_{2} |}{e^{C + t}}$

Step 7.5.4.2.2

Simplify the left side.

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Step 7.5.4.2.2.1

Cancel the common factor of $e^{C + t}$ .

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Step 7.5.4.2.2.1.1

Cancel the common factor.

$\frac{e^{C + t} | v |}{e^{C + t}} = \frac{| u_{2} |}{e^{C + t}}$

Step 7.5.4.2.2.1.2

Divide $| v |$ by $1$ .

$| v | = \frac{| u_{2} |}{e^{C + t}}$

Step 7.5.4.3

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

$v = \pm \frac{| u_{2} |}{e^{C + t}}$

Step 8

Group the constant terms together.

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Step 8.1

Reorder terms.

$v = \pm \frac{| u_{2} |}{e^{t + C}}$

Step 8.2

Rewrite $e^{t + C}$ as $e^{t} e^{C}$ .

$v = \pm \frac{| u_{2} |}{e^{t} e^{C}}$

Step 8.3

Reorder $e^{t}$ and $e^{C}$ .

$v = \pm \frac{| u_{2} |}{e^{C} e^{t}}$

Step 9

Replace all occurrences of $v$ with $e^{x - t}$ .

$e^{x - t} = \pm \frac{| u_{2} |}{e^{C} e^{t}}$

Step 10

Solve for $x$ .

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Step 10.1

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

$\ln (e^{x - t}) = \ln (\pm \frac{| u_{2} |}{e^{C} e^{t}})$

Step 10.2

Expand the left side.

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Step 10.2.1

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

$(x - t) \ln (e) = \ln (\pm \frac{| u_{2} |}{e^{C} e^{t}})$

Step 10.2.2

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

$(x - t) \cdot 1 = \ln (\pm \frac{| u_{2} |}{e^{C} e^{t}})$

Step 10.2.3

Multiply $x - t$ by $1$ .

$x - t = \ln (\pm \frac{| u_{2} |}{e^{C} e^{t}})$

Step 10.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x - t = \ln (\pm \frac{| u_{2} |}{e^{C + t}})$

Step 10.4

Add $t$ to both sides of the equation.

$x = \ln (\pm \frac{| u_{2} |}{e^{C + t}}) + t$

Step 11

Group the constant terms together.

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Step 11.1

Reorder terms.

$x = \ln (\pm \frac{| u_{2} |}{e^{t + C}}) + t$

Step 11.2

Rewrite $e^{t + C}$ as $e^{t} e^{C}$ .

$x = \ln (\pm \frac{| u_{2} |}{e^{t} e^{C}}) + t$

Step 11.3

Reorder $e^{t}$ and $e^{C}$ .

$x = \ln (\pm \frac{| u_{2} |}{e^{C} e^{t}}) + t$