Calculus Examples

\frac{d y}{d x} = e^{- 11 x}

$\frac{d y}{d x} = e^{- 11 x}$ ,

(0, 13)

$(0, 13)$

Step 1

Rewrite the equation.

d y = e^{- 11 x} d x

$d y = e^{- 11 x} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

\int d y = \int e^{- 11 x} d x

$\int d y = \int e^{- 11 x} d x$

Step 2.2

Apply the constant rule.

y + C_{1} = \int e^{- 11 x} d x

$y + C_{1} = \int e^{- 11 x} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let

u = - 11 x

$u = - 11 x$ . Then

d u = - 11 d x

$d u = - 11 d x$ , so

- \frac{1}{11} d u = d x

$- \frac{1}{11} d u = d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 2.3.1.1

Let

u = - 11 x

$u = - 11 x$ . Find

\frac{d u}{d x}

$\frac{d u}{d x}$ .

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Step 2.3.1.1.1

Differentiate

- 11 x

$- 11 x$ .

\frac{d}{d x} [- 11 x]

$\frac{d}{d x} [- 11 x]$

Step 2.3.1.1.2

Since

- 11

$- 11$ is constant with respect to

x

$x$ , the derivative of

- 11 x

$- 11 x$ with respect to

x

$x$ is

- 11 \frac{d}{d x} [x]

$- 11 \frac{d}{d x} [x]$ .

- 11 \frac{d}{d x} [x]

$- 11 \frac{d}{d x} [x]$

Step 2.3.1.1.3

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 1

$n = 1$ .

- 11 \cdot 1

$- 11 \cdot 1$

Step 2.3.1.1.4

Multiply

- 11

$- 11$ by

1

$1$ .

- 11

$- 11$

- 11

$- 11$

Step 2.3.1.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

y + C_{1} = \int e^{u} \frac{1}{- 11} d u

$y + C_{1} = \int e^{u} \frac{1}{- 11} d u$

y + C_{1} = \int e^{u} \frac{1}{- 11} d u

$y + C_{1} = \int e^{u} \frac{1}{- 11} d u$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Move the negative in front of the fraction.

y + C_{1} = \int e^{u} (- \frac{1}{11}) d u

$y + C_{1} = \int e^{u} (- \frac{1}{11}) d u$

Step 2.3.2.2

Combine

e^{u}

$e^{u}$ and

\frac{1}{11}

$\frac{1}{11}$ .

y + C_{1} = \int - \frac{e^{u}}{11} d u

$y + C_{1} = \int - \frac{e^{u}}{11} d u$

y + C_{1} = \int - \frac{e^{u}}{11} d u

$y + C_{1} = \int - \frac{e^{u}}{11} d u$

Step 2.3.3

Since

- 1

$- 1$ is constant with respect to

u

$u$ , move

- 1

$- 1$ out of the integral.

y + C_{1} = - \int \frac{e^{u}}{11} d u

$y + C_{1} = - \int \frac{e^{u}}{11} d u$

Step 2.3.4

Since

\frac{1}{11}

$\frac{1}{11}$ is constant with respect to

u

$u$ , move

\frac{1}{11}

$\frac{1}{11}$ out of the integral.

y + C_{1} = - (\frac{1}{11} \int e^{u} d u)

$y + C_{1} = - (\frac{1}{11} \int e^{u} d u)$

Step 2.3.5

The integral of

e^{u}

$e^{u}$ with respect to

u

$u$ is

e^{u}

$e^{u}$ .

y + C_{1} = - \frac{1}{11} (e^{u} + C_{2})

$y + C_{1} = - \frac{1}{11} (e^{u} + C_{2})$

Step 2.3.6

Simplify.

y + C_{1} = - \frac{1}{11} e^{u} + C_{2}

$y + C_{1} = - \frac{1}{11} e^{u} + C_{2}$

Step 2.3.7

Replace all occurrences of

u

$u$ with

- 11 x

$- 11 x$ .

y + C_{1} = - \frac{1}{11} e^{- 11 x} + C_{2}

$y + C_{1} = - \frac{1}{11} e^{- 11 x} + C_{2}$

y + C_{1} = - \frac{1}{11} e^{- 11 x} + C_{2}

$y + C_{1} = - \frac{1}{11} e^{- 11 x} + C_{2}$

Step 2.4

Group the constant of integration on the right side as

K

$K$ .

y = - \frac{1}{11} e^{- 11 x} + K

$y = - \frac{1}{11} e^{- 11 x} + K$

y = - \frac{1}{11} e^{- 11 x} + K

$y = - \frac{1}{11} e^{- 11 x} + K$

Step 3

Use the initial condition to find the value of

K

$K$ by substituting

0

$0$ for

x

$x$ and

13

$13$ for

y

$y$ in

y = - \frac{1}{11} e^{- 11 x} + K

$y = - \frac{1}{11} e^{- 11 x} + K$ .

13 = - \frac{1}{11} e^{- 11 \cdot 0} + K

$13 = - \frac{1}{11} e^{- 11 \cdot 0} + K$

Step 4

Solve for

K

$K$ .

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Step 4.1

Rewrite the equation as

- \frac{1}{11} \cdot e^{- 11 \cdot 0} + K = 13

$- \frac{1}{11} \cdot e^{- 11 \cdot 0} + K = 13$ .

- \frac{1}{11} \cdot e^{- 11 \cdot 0} + K = 13

$- \frac{1}{11} \cdot e^{- 11 \cdot 0} + K = 13$

Step 4.2

Simplify each term.

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Step 4.2.1

Multiply

- 11

$- 11$ by

0

$0$ .

- \frac{1}{11} \cdot e^{0} + K = 13

$- \frac{1}{11} \cdot e^{0} + K = 13$

Step 4.2.2

Anything raised to

0

$0$ is

1

$1$ .

- \frac{1}{11} \cdot 1 + K = 13

$- \frac{1}{11} \cdot 1 + K = 13$

Step 4.2.3

Multiply

- 1

$- 1$ by

1

$1$ .

- \frac{1}{11} + K = 13

$- \frac{1}{11} + K = 13$

- \frac{1}{11} + K = 13

$- \frac{1}{11} + K = 13$

Step 4.3

Move all terms not containing

K

$K$ to the right side of the equation.

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Step 4.3.1

Add

\frac{1}{11}

$\frac{1}{11}$ to both sides of the equation.

K = 13 + \frac{1}{11}

$K = 13 + \frac{1}{11}$

Step 4.3.2

To write

13

$13$ as a fraction with a common denominator, multiply by

\frac{11}{11}

$\frac{11}{11}$ .

K = 13 \cdot \frac{11}{11} + \frac{1}{11}

$K = 13 \cdot \frac{11}{11} + \frac{1}{11}$

Step 4.3.3

Combine

13

$13$ and

\frac{11}{11}

$\frac{11}{11}$ .

K = \frac{13 \cdot 11}{11} + \frac{1}{11}

$K = \frac{13 \cdot 11}{11} + \frac{1}{11}$

Step 4.3.4

Combine the numerators over the common denominator.

K = \frac{13 \cdot 11 + 1}{11}

$K = \frac{13 \cdot 11 + 1}{11}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply

13

$13$ by

11

$11$ .

K = \frac{143 + 1}{11}

$K = \frac{143 + 1}{11}$

Step 4.3.5.2

Add

143

$143$ and

1

$1$ .

K = \frac{144}{11}

$K = \frac{144}{11}$

K = \frac{144}{11}

$K = \frac{144}{11}$

K = \frac{144}{11}

$K = \frac{144}{11}$

K = \frac{144}{11}

$K = \frac{144}{11}$

Step 5

Substitute

\frac{144}{11}

$\frac{144}{11}$ for

K

$K$ in

y = - \frac{1}{11} e^{- 11 x} + K

$y = - \frac{1}{11} e^{- 11 x} + K$ and simplify.

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Step 5.1

Substitute

\frac{144}{11}

$\frac{144}{11}$ for

K

$K$ .

y = - \frac{1}{11} e^{- 11 x} + \frac{144}{11}

$y = - \frac{1}{11} e^{- 11 x} + \frac{144}{11}$

Step 5.2

Combine

e^{- 11 x}

$e^{- 11 x}$ and

\frac{1}{11}

$\frac{1}{11}$ .

y = - \frac{e^{- 11 x}}{11} + \frac{144}{11}

$y = - \frac{e^{- 11 x}}{11} + \frac{144}{11}$

y = - \frac{e^{- 11 x}}{11} + \frac{144}{11}

$y = - \frac{e^{- 11 x}}{11} + \frac{144}{11}$

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