Calculus Examples

$\frac{d y}{d x} = \frac{- 7 y + 4}{6 x - 7}$

Step 1

Separate the variables.

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Step 1.1

Multiply both sides by $\frac{1}{- 7 y + 4}$ .

$\frac{1}{- 7 y + 4} \frac{d y}{d x} = \frac{1}{- 7 y + 4} \cdot \frac{- 7 y + 4}{6 x - 7}$

Step 1.2

Simplify.

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Step 1.2.1

Multiply $\frac{1}{- 7 y + 4}$ by $\frac{- 7 y + 4}{6 x - 7}$ .

$\frac{1}{- 7 y + 4} \frac{d y}{d x} = \frac{- 7 y + 4}{(- 7 y + 4) (6 x - 7)}$

Step 1.2.2

Cancel the common factor of $- 7 y + 4$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{1}{- 7 y + 4} \frac{d y}{d x} = \frac{- 7 y + 4}{(- 7 y + 4) (6 x - 7)}$

Step 1.2.2.2

Rewrite the expression.

$\frac{1}{- 7 y + 4} \frac{d y}{d x} = \frac{1}{6 x - 7}$

Step 1.3

Rewrite the equation.

$\frac{1}{- 7 y + 4} d y = \frac{1}{6 x - 7} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{- 7 y + 4} d y = \int \frac{1}{6 x - 7} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Let $u_{1} = - 7 y + 4$ . Then $d u_{1} = - 7 d y$ , so $- \frac{1}{7} d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 2.2.1.1

Let $u_{1} = - 7 y + 4$ . Find $\frac{d u_{1}}{d y}$ .

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Step 2.2.1.1.1

Differentiate $- 7 y + 4$ .

$\frac{d}{d y} [- 7 y + 4]$

Step 2.2.1.1.2

By the Sum Rule, the derivative of $- 7 y + 4$ with respect to $y$ is $\frac{d}{d y} [- 7 y] + \frac{d}{d y} [4]$ .

$\frac{d}{d y} [- 7 y] + \frac{d}{d y} [4]$

Step 2.2.1.1.3

Evaluate $\frac{d}{d y} [- 7 y]$ .

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Step 2.2.1.1.3.1

Since $- 7$ is constant with respect to $y$ , the derivative of $- 7 y$ with respect to $y$ is $- 7 \frac{d}{d y} [y]$ .

$- 7 \frac{d}{d y} [y] + \frac{d}{d y} [4]$

Step 2.2.1.1.3.2

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

$- 7 \cdot 1 + \frac{d}{d y} [4]$

Step 2.2.1.1.3.3

Multiply $- 7$ by $1$ .

$- 7 + \frac{d}{d y} [4]$

Step 2.2.1.1.4

Differentiate using the Constant Rule.

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Step 2.2.1.1.4.1

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

$- 7 + 0$

Step 2.2.1.1.4.2

Add $- 7$ and $0$ .

$- 7$

Step 2.2.1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{1}{u_{1}} \cdot \frac{1}{- 7} d u_{1} = \int \frac{1}{6 x - 7} d x$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Move the negative in front of the fraction.

$\int \frac{1}{u_{1}} (- \frac{1}{7}) d u_{1} = \int \frac{1}{6 x - 7} d x$

Step 2.2.2.2

Multiply $\frac{1}{u_{1}}$ by $\frac{1}{7}$ .

$\int - \frac{1}{u_{1} \cdot 7} d u_{1} = \int \frac{1}{6 x - 7} d x$

Step 2.2.2.3

Move $7$ to the left of $u_{1}$ .

$\int - \frac{1}{7 u_{1}} d u_{1} = \int \frac{1}{6 x - 7} d x$

Step 2.2.3

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

$- \int \frac{1}{7 u_{1}} d u_{1} = \int \frac{1}{6 x - 7} d x$

Step 2.2.4

Since $\frac{1}{7}$ is constant with respect to $u_{1}$ , move $\frac{1}{7}$ out of the integral.

$- (\frac{1}{7} \int \frac{1}{u_{1}} d u_{1}) = \int \frac{1}{6 x - 7} d x$

Step 2.2.5

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

$- \frac{1}{7} (\ln (| u_{1} |) + C_{1}) = \int \frac{1}{6 x - 7} d x$

Step 2.2.6

Simplify.

$- \frac{1}{7} \ln (| u_{1} |) + C_{1} = \int \frac{1}{6 x - 7} d x$

Step 2.2.7

Replace all occurrences of $u_{1}$ with $- 7 y + 4$ .

$- \frac{1}{7} \ln (| - 7 y + 4 |) + C_{1} = \int \frac{1}{6 x - 7} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

Let $u_{2} = 6 x - 7$ . Then $d u_{2} = 6 d x$ , so $\frac{1}{6} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.3.1.1

Let $u_{2} = 6 x - 7$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.3.1.1.1

Differentiate $6 x - 7$ .

$\frac{d}{d x} [6 x - 7]$

Step 2.3.1.1.2

By the Sum Rule, the derivative of $6 x - 7$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 7]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [- 7]$

Step 2.3.1.1.3

Evaluate $\frac{d}{d x} [6 x]$ .

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Step 2.3.1.1.3.1

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

$6 \frac{d}{d x} [x] + \frac{d}{d x} [- 7]$

Step 2.3.1.1.3.2

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

$6 \cdot 1 + \frac{d}{d x} [- 7]$

Step 2.3.1.1.3.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [- 7]$

Step 2.3.1.1.4

Differentiate using the Constant Rule.

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Step 2.3.1.1.4.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$6 + 0$

Step 2.3.1.1.4.2

Add $6$ and $0$ .

$6$

Step 2.3.1.2

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

$- \frac{1}{7} \ln (| - 7 y + 4 |) + C_{1} = \int \frac{1}{u_{2}} \cdot \frac{1}{6} d u_{2}$

Step 2.3.2

Simplify.

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Step 2.3.2.1

Multiply $\frac{1}{u_{2}}$ by $\frac{1}{6}$ .

$- \frac{1}{7} \ln (| - 7 y + 4 |) + C_{1} = \int \frac{1}{u_{2} \cdot 6} d u_{2}$

Step 2.3.2.2

Move $6$ to the left of $u_{2}$ .

$- \frac{1}{7} \ln (| - 7 y + 4 |) + C_{1} = \int \frac{1}{6 u_{2}} d u_{2}$

Step 2.3.3

Since $\frac{1}{6}$ is constant with respect to $u_{2}$ , move $\frac{1}{6}$ out of the integral.

$- \frac{1}{7} \ln (| - 7 y + 4 |) + C_{1} = \frac{1}{6} \int \frac{1}{u_{2}} d u_{2}$

Step 2.3.4

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

$- \frac{1}{7} \ln (| - 7 y + 4 |) + C_{1} = \frac{1}{6} (\ln (| u_{2} |) + C_{2})$

Step 2.3.5

Simplify.

$- \frac{1}{7} \ln (| - 7 y + 4 |) + C_{1} = \frac{1}{6} \ln (| u_{2} |) + C_{2}$

Step 2.3.6

Replace all occurrences of $u_{2}$ with $6 x - 7$ .

$- \frac{1}{7} \ln (| - 7 y + 4 |) + C_{1} = \frac{1}{6} \ln (| 6 x - 7 |) + C_{2}$

Step 2.4

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

$- \frac{1}{7} \ln (| - 7 y + 4 |) = \frac{1}{6} \ln (| 6 x - 7 |) + C$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $- 7$ .

$- 7 (- \frac{1}{7} \ln (| - 7 y + 4 |)) = - 7 (\frac{1}{6} \ln (| 6 x - 7 |) + C)$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $- 7 (- \frac{1}{7} \ln (| - 7 y + 4 |))$ .

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Step 3.2.1.1.1

Combine $\ln (| - 7 y + 4 |)$ and $\frac{1}{7}$ .

$- 7 (- \frac{\ln (| - 7 y + 4 |)}{7}) = - 7 (\frac{1}{6} \ln (| 6 x - 7 |) + C)$

Step 3.2.1.1.2

Cancel the common factor of $7$ .

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Step 3.2.1.1.2.1

Move the leading negative in $- \frac{\ln (| - 7 y + 4 |)}{7}$ into the numerator.

$- 7 \frac{- \ln (| - 7 y + 4 |)}{7} = - 7 (\frac{1}{6} \ln (| 6 x - 7 |) + C)$

Step 3.2.1.1.2.2

Factor $7$ out of $- 7$ .

$7 (- 1) \frac{- \ln (| - 7 y + 4 |)}{7} = - 7 (\frac{1}{6} \ln (| 6 x - 7 |) + C)$

Step 3.2.1.1.2.3

Cancel the common factor.

$7 \cdot - 1 \frac{- \ln (| - 7 y + 4 |)}{7} = - 7 (\frac{1}{6} \ln (| 6 x - 7 |) + C)$

Step 3.2.1.1.2.4

Rewrite the expression.

$- - \ln (| - 7 y + 4 |) = - 7 (\frac{1}{6} \ln (| 6 x - 7 |) + C)$

Step 3.2.1.1.3

Multiply.

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Step 3.2.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \ln (| - 7 y + 4 |) = - 7 (\frac{1}{6} \ln (| 6 x - 7 |) + C)$

Step 3.2.1.1.3.2

Multiply $\ln (| - 7 y + 4 |)$ by $1$ .

$\ln (| - 7 y + 4 |) = - 7 (\frac{1}{6} \ln (| 6 x - 7 |) + C)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $- 7 (\frac{1}{6} \ln (| 6 x - 7 |) + C)$ .

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Step 3.2.2.1.1

Combine $\frac{1}{6}$ and $\ln (| 6 x - 7 |)$ .

$\ln (| - 7 y + 4 |) = - 7 (\frac{\ln (| 6 x - 7 |)}{6} + C)$

Step 3.2.2.1.2

To write $C$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\ln (| - 7 y + 4 |) = - 7 (\frac{\ln (| 6 x - 7 |)}{6} + C \cdot \frac{6}{6})$

Step 3.2.2.1.3

Simplify terms.

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Step 3.2.2.1.3.1

Combine $C$ and $\frac{6}{6}$ .

$\ln (| - 7 y + 4 |) = - 7 (\frac{\ln (| 6 x - 7 |)}{6} + \frac{C \cdot 6}{6})$

Step 3.2.2.1.3.2

Combine the numerators over the common denominator.

$\ln (| - 7 y + 4 |) = - 7 \frac{\ln (| 6 x - 7 |) + C \cdot 6}{6}$

Step 3.2.2.1.4

Move $6$ to the left of $C$ .

$\ln (| - 7 y + 4 |) = - 7 \frac{\ln (| 6 x - 7 |) + 6 C}{6}$

Step 3.2.2.1.5

Combine fractions.

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Step 3.2.2.1.5.1

Combine $- 7$ and $\frac{\ln (| 6 x - 7 |) + 6 C}{6}$ .

$\ln (| - 7 y + 4 |) = \frac{- 7 (\ln (| 6 x - 7 |) + 6 C)}{6}$

Step 3.2.2.1.5.2

Move the negative in front of the fraction.

$\ln (| - 7 y + 4 |) = - \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}$

Step 3.3

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

$e^{\ln (| - 7 y + 4 |)} = e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}}$

Step 3.4

Rewrite $\ln (| - 7 y + 4 |) = - \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}$ 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 $b^{y} = x$ .

$e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}} = | - 7 y + 4 |$

Step 3.5

Solve for $y$ .

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Step 3.5.1

Rewrite the equation as $| - 7 y + 4 | = e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}}$ .

$| - 7 y + 4 | = e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}}$

Step 3.5.2

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

$- 7 y + 4 = \pm e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}}$

Step 3.5.3

Subtract $4$ from both sides of the equation.

$- 7 y = \pm e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}} - 4$

Step 3.5.4

Divide each term in $- 7 y = \pm e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}} - 4$ by $- 7$ and simplify.

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Step 3.5.4.1

Divide each term in $- 7 y = \pm e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}} - 4$ by $- 7$ .

$\frac{- 7 y}{- 7} = \frac{\pm e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}}}{- 7} + \frac{- 4}{- 7}$

Step 3.5.4.2

Simplify the left side.

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Step 3.5.4.2.1

Cancel the common factor of $- 7$ .

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Step 3.5.4.2.1.1

Cancel the common factor.

$\frac{- 7 y}{- 7} = \frac{\pm e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}}}{- 7} + \frac{- 4}{- 7}$

Step 3.5.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\pm e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}}}{- 7} + \frac{- 4}{- 7}$

Step 3.5.4.3

Simplify the right side.

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Step 3.5.4.3.1

Simplify each term.

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Step 3.5.4.3.1.1

Simplify $\frac{\pm e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}}}{- 7}$ .

$y = \frac{\pm e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}}}{7} + \frac{- 4}{- 7}$

Step 3.5.4.3.1.2

Dividing two negative values results in a positive value.

$y = \frac{\pm e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}}}{7} + \frac{4}{7}$

Step 3.5.4.3.2

Combine the numerators over the common denominator.

$y = \frac{\pm e^{- \frac{7 (\ln (| 6 x - 7 |) + 6 C)}{6}} + 4}{7}$

Step 4

Simplify the constant of integration.

$y = \frac{\pm e^{- \frac{7 (\ln (| 6 x - 7 |) + C)}{6}} + 4}{7}$