Calculus Examples

$e^{- y} \sec (x) - \frac{d y}{d x} \cos (x) = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Simplify the left side.

$e^{- y} \sec (x) - \frac{d y}{d x} \cos (x) = 0$

Step 1.1.2

Subtract $e^{- y} \sec (x)$ from both sides of the equation.

$- \frac{d y}{d x} \cos (x) = - e^{- y} \sec (x)$

Step 1.1.3

Divide each term in $- \frac{d y}{d x} \cos (x) = - e^{- y} \sec (x)$ by $- \frac{d y}{d x} \cos (x)$ and simplify.

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Step 1.1.3.1

Divide each term in $- \frac{d y}{d x} \cos (x) = - e^{- y} \sec (x)$ by $- \frac{d y}{d x} \cos (x)$ .

$\frac{- \frac{d y}{d x} \cos (x)}{- \frac{d y}{d x} \cos (x)} = \frac{- e^{- y} \sec (x)}{- \frac{d y}{d x} \cos (x)}$

Step 1.1.3.2

Simplify the left side.

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Step 1.1.3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{d y}{d x} \cos (x)}{\frac{d y}{d x} \cos (x)} = \frac{- e^{- y} \sec (x)}{- \frac{d y}{d x} \cos (x)}$

Step 1.1.3.2.2

Cancel the common factor of $\frac{d y}{d x}$ .

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Step 1.1.3.2.2.1

Cancel the common factor.

$\frac{\frac{d y}{d x} \cos (x)}{\frac{d y}{d x} \cos (x)} = \frac{- e^{- y} \sec (x)}{- \frac{d y}{d x} \cos (x)}$

Step 1.1.3.2.2.2

Rewrite the expression.

$\frac{\cos (x)}{\cos (x)} = \frac{- e^{- y} \sec (x)}{- \frac{d y}{d x} \cos (x)}$

Step 1.1.3.2.3

Cancel the common factor of $\cos (x)$ .

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Step 1.1.3.2.3.1

Cancel the common factor.

$\frac{\cos (x)}{\cos (x)} = \frac{- e^{- y} \sec (x)}{- \frac{d y}{d x} \cos (x)}$

Step 1.1.3.2.3.2

Rewrite the expression.

$1 = \frac{- e^{- y} \sec (x)}{- \frac{d y}{d x} \cos (x)}$

Step 1.1.3.3

Simplify the right side.

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Step 1.1.3.3.1

Dividing two negative values results in a positive value.

$1 = \frac{e^{- y} \sec (x)}{\frac{d y}{d x} \cos (x)}$

Step 1.1.3.3.2

Separate fractions.

$1 = \frac{e^{- y}}{\frac{d y}{d x}} \cdot \frac{\sec (x)}{\cos (x)}$

Step 1.1.3.3.3

Rewrite $\sec (x)$ in terms of sines and cosines.

$1 = \frac{e^{- y}}{\frac{d y}{d x}} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)}$

Step 1.1.3.3.4

Rewrite $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ as a product.

$1 = \frac{e^{- y}}{\frac{d y}{d x}} (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)})$

Step 1.1.3.3.5

Combine fractions.

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Step 1.1.3.3.5.1

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

$1 = \frac{e^{- y}}{\frac{d y}{d x}} \cdot \frac{1}{\cos (x) \cos (x)}$

Step 1.1.3.3.5.2

Combine.

$1 = \frac{e^{- y} \cdot 1}{\frac{d y}{d x} (\cos (x) \cos (x))}$

Step 1.1.3.3.5.3

Multiply $e^{- y}$ by $1$ .

$1 = \frac{e^{- y}}{\frac{d y}{d x} \cos (x) \cos (x)}$

Step 1.1.3.3.6

Simplify the denominator.

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Step 1.1.3.3.6.1

Raise $\cos (x)$ to the power of $1$ .

$1 = \frac{e^{- y}}{\frac{d y}{d x} (\cos^{1} (x) \cos (x))}$

Step 1.1.3.3.6.2

Raise $\cos (x)$ to the power of $1$ .

$1 = \frac{e^{- y}}{\frac{d y}{d x} (\cos^{1} (x) \cos^{1} (x))}$

Step 1.1.3.3.6.3

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

$1 = \frac{e^{- y}}{\frac{d y}{d x} {\cos (x)}^{1 + 1}}$

Step 1.1.3.3.6.4

Add $1$ and $1$ .

$1 = \frac{e^{- y}}{\frac{d y}{d x} \cos^{2} (x)}$

Step 1.1.3.3.7

Multiply by $1$ .

$1 = \frac{e^{- y}}{\frac{d y}{d x} (\cos^{2} (x) \cdot 1)}$

Step 1.1.3.3.8

Separate fractions.

$1 = \frac{e^{- y}}{\frac{d y}{d x} \cdot 1} \cdot \frac{1}{\cos^{2} (x)}$

Step 1.1.3.3.9

Convert from $\frac{1}{\cos^{2} (x)}$ to $\sec^{2} (x)$ .

$1 = \frac{e^{- y}}{\frac{d y}{d x} \cdot 1} \sec^{2} (x)$

Step 1.1.3.3.10

Multiply $\frac{d y}{d x}$ by $1$ .

$1 = \frac{e^{- y}}{\frac{d y}{d x}} \sec^{2} (x)$

Step 1.1.3.3.11

Combine $\frac{e^{- y}}{\frac{d y}{d x}}$ and $\sec^{2} (x)$ .

$1 = \frac{e^{- y} \sec^{2} (x)}{\frac{d y}{d x}}$

Step 1.1.4

Rewrite the equation as $\frac{e^{- y} \sec^{2} (x)}{\frac{d y}{d x}} = 1$ .

$\frac{e^{- y} \sec^{2} (x)}{\frac{d y}{d x}} = 1$

Step 1.1.5

Find the LCD of the terms in the equation.

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Step 1.1.5.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$\frac{d y}{d x}, 1$

Step 1.1.5.2

The LCM of one and any expression is the expression.

$\frac{d y}{d x}$

Step 1.1.6

Multiply each term in $\frac{e^{- y} \sec^{2} (x)}{\frac{d y}{d x}} = 1$ by $\frac{d y}{d x}$ to eliminate the fractions.

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Step 1.1.6.1

Multiply each term in $\frac{e^{- y} \sec^{2} (x)}{\frac{d y}{d x}} = 1$ by $\frac{d y}{d x}$ .

$\frac{e^{- y} \sec^{2} (x)}{\frac{d y}{d x}} \frac{d y}{d x} = 1 \frac{d y}{d x}$

Step 1.1.6.2

Simplify the left side.

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Step 1.1.6.2.1

Cancel the common factor of $\frac{d y}{d x}$ .

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Step 1.1.6.2.1.1

Cancel the common factor.

$\frac{e^{- y} \sec^{2} (x)}{\frac{d y}{d x}} \frac{d y}{d x} = 1 \frac{d y}{d x}$

Step 1.1.6.2.1.2

Rewrite the expression.

$e^{- y} \sec^{2} (x) = 1 \frac{d y}{d x}$

Step 1.1.6.3

Simplify the right side.

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Step 1.1.6.3.1

Multiply $\frac{d y}{d x}$ by $1$ .

$e^{- y} \sec^{2} (x) = \frac{d y}{d x}$

Step 1.1.7

Rewrite the equation as $\frac{d y}{d x} = e^{- y} \sec^{2} (x)$ .

$\frac{d y}{d x} = e^{- y} \sec^{2} (x)$

Step 1.2

Multiply both sides by $\frac{1}{e^{- y}}$ .

$\frac{1}{e^{- y}} \frac{d y}{d x} = \frac{1}{e^{- y}} (e^{- y} \sec^{2} (x))$

Step 1.3

Cancel the common factor of $e^{- y}$ .

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Step 1.3.1

Factor $e^{- y}$ out of $e^{- y} \sec^{2} (x)$ .

$\frac{1}{e^{- y}} \frac{d y}{d x} = \frac{1}{e^{- y}} (e^{- y} (\sec^{2} (x)))$

Step 1.3.2

Cancel the common factor.

$\frac{1}{e^{- y}} \frac{d y}{d x} = \frac{1}{e^{- y}} (e^{- y} \sec^{2} (x))$

Step 1.3.3

Rewrite the expression.

$\frac{1}{e^{- y}} \frac{d y}{d x} = \sec^{2} (x)$

Step 1.4

Rewrite the equation.

$\frac{1}{e^{- y}} d y = \sec^{2} (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{1}{e^{- y}} d y = \int \sec^{2} (x) d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

Negate the exponent of $e^{- y}$ and move it out of the denominator.

$\int 1 {(e^{- y})}^{- 1} d y = \int \sec^{2} (x) d x$

Step 2.2.1.2

Simplify.

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Step 2.2.1.2.1

Multiply the exponents in ${(e^{- y})}^{- 1}$ .

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Step 2.2.1.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int 1 e^{- y \cdot - 1} d y = \int \sec^{2} (x) d x$

Step 2.2.1.2.1.2

Multiply $- y \cdot - 1$ .

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Step 2.2.1.2.1.2.1

Multiply $- 1$ by $- 1$ .

$\int 1 e^{1 y} d y = \int \sec^{2} (x) d x$

Step 2.2.1.2.1.2.2

Multiply $y$ by $1$ .

$\int 1 e^{y} d y = \int \sec^{2} (x) d x$

Step 2.2.1.2.2

Multiply $e^{y}$ by $1$ .

$\int e^{y} d y = \int \sec^{2} (x) d x$

Step 2.2.2

The integral of $e^{y}$ with respect to $y$ is $e^{y}$ .

$e^{y} + C_{1} = \int \sec^{2} (x) d x$

Step 2.3

Since the derivative of $\tan (x)$ is $\sec^{2} (x)$ , the integral of $\sec^{2} (x)$ is $\tan (x)$ .

$e^{y} + C_{1} = \tan (x) + C_{2}$

Step 2.4

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

$e^{y} = \tan (x) + K$

Step 3

Solve for $y$ .

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Step 3.1

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

$\ln (e^{y}) = \ln (\tan (x) + K)$

Step 3.2

Expand the left side.

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Step 3.2.1

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

$y \ln (e) = \ln (\tan (x) + K)$

Step 3.2.2

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

$y \cdot 1 = \ln (\tan (x) + K)$

Step 3.2.3

Multiply $y$ by $1$ .

$y = \ln (\tan (x) + K)$