Calculus Examples

$y = \frac{e^{y}}{1 + \sin (x)}$

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = e^{y}$ and $g (x) = 1 + \sin (x)$ .

$\frac{(1 + \sin (x)) \frac{d}{d x} [e^{y}] - e^{y} \frac{d}{d x} [1 + \sin (x)]}{{(1 + \sin (x))}^{2}}$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = y$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $y$ .

$\frac{(1 + \sin (x)) (\frac{d}{d u} [e^{u}] \frac{d}{d x} [y]) - e^{y} \frac{d}{d x} [1 + \sin (x)]}{{(1 + \sin (x))}^{2}}$

Step 3.2.2

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

$\frac{(1 + \sin (x)) (e^{u} \frac{d}{d x} [y]) - e^{y} \frac{d}{d x} [1 + \sin (x)]}{{(1 + \sin (x))}^{2}}$

Step 3.2.3

Replace all occurrences of $u$ with $y$ .

$\frac{(1 + \sin (x)) (e^{y} \frac{d}{d x} [y]) - e^{y} \frac{d}{d x} [1 + \sin (x)]}{{(1 + \sin (x))}^{2}}$

Step 3.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\frac{(1 + \sin (x)) e^{y} y' - e^{y} \frac{d}{d x} [1 + \sin (x)]}{{(1 + \sin (x))}^{2}}$

Step 3.4

Differentiate.

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Step 3.4.1

By the Sum Rule, the derivative of $1 + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\sin (x)]$ .

$\frac{(1 + \sin (x)) e^{y} y' - e^{y} (\frac{d}{d x} [1] + \frac{d}{d x} [\sin (x)])}{{(1 + \sin (x))}^{2}}$

Step 3.4.2

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

$\frac{(1 + \sin (x)) e^{y} y' - e^{y} (0 + \frac{d}{d x} [\sin (x)])}{{(1 + \sin (x))}^{2}}$

Step 3.4.3

Add $0$ and $\frac{d}{d x} [\sin (x)]$ .

$\frac{(1 + \sin (x)) e^{y} y' - e^{y} \frac{d}{d x} [\sin (x)]}{{(1 + \sin (x))}^{2}}$

Step 3.5

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\frac{(1 + \sin (x)) e^{y} y' - e^{y} \cos (x)}{{(1 + \sin (x))}^{2}}$

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

$\frac{(1 e^{y} + \sin (x) e^{y}) y' - e^{y} \cos (x)}{{(1 + \sin (x))}^{2}}$

Step 3.6.2

Apply the distributive property.

$\frac{1 e^{y} y' + \sin (x) e^{y} y' - e^{y} \cos (x)}{{(1 + \sin (x))}^{2}}$

Step 3.6.3

Simplify the numerator.

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Step 3.6.3.1

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

$\frac{e^{y} y' + \sin (x) e^{y} y' - e^{y} \cos (x)}{{(1 + \sin (x))}^{2}}$

Step 3.6.3.2

Reorder factors in $e^{y} y' + \sin (x) e^{y} y' - e^{y} \cos (x)$ .

$\frac{e^{y} y' + e^{y} \sin (x) y' - e^{y} \cos (x)}{{(1 + \sin (x))}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

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

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by ${(1 + \sin (x))}^{2}$ .

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{e^{y} y' + e^{y} \sin (x) y' - e^{y} \cos (x)}{{(1 + \sin (x))}^{2}} {(1 + \sin (x))}^{2}$ .

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Step 5.2.1.1

Cancel the common factor of ${(1 + \sin (x))}^{2}$ .

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Step 5.2.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.2

Rewrite the expression.

$y' {(1 + \sin (x))}^{2} = e^{y} y' + e^{y} \sin (x) y' - e^{y} \cos (x)$

Step 5.2.1.2

Reorder factors in $e^{y} y' + e^{y} \sin (x) y' - e^{y} \cos (x)$ .

$y' {(1 + \sin (x))}^{2} = y' e^{y} + y' e^{y} \sin (x) - e^{y} \cos (x)$

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$y' e^{y} + y' e^{y} \sin (x) - e^{y} \cos (x) = y' {(1 + \sin (x))}^{2}$

Step 5.3.2

Simplify $y' {(1 + \sin (x))}^{2}$ .

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Step 5.3.2.1

Rewrite.

$y' e^{y} + y' e^{y} \sin (x) - e^{y} \cos (x) = 0 + 0 + y' {(1 + \sin (x))}^{2}$

Step 5.3.2.2

Rewrite ${(1 + \sin (x))}^{2}$ as $(1 + \sin (x)) (1 + \sin (x))$ .

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' ((1 + \sin (x)) (1 + \sin (x)))$

Step 5.3.2.3

Expand $(1 + \sin (x)) (1 + \sin (x))$ using the FOIL Method.

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Step 5.3.2.3.1

Apply the distributive property.

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' (1 (1 + \sin (x)) + \sin (x) (1 + \sin (x)))$

Step 5.3.2.3.2

Apply the distributive property.

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' (1 \cdot 1 + 1 \sin (x) + \sin (x) (1 + \sin (x)))$

Step 5.3.2.3.3

Apply the distributive property.

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' (1 \cdot 1 + 1 \sin (x) + \sin (x) \cdot 1 + \sin (x) \sin (x))$

Step 5.3.2.4

Simplify and combine like terms.

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Step 5.3.2.4.1

Simplify each term.

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Step 5.3.2.4.1.1

Multiply $1$ by $1$ .

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' (1 + 1 \sin (x) + \sin (x) \cdot 1 + \sin (x) \sin (x))$

Step 5.3.2.4.1.2

Multiply $\sin (x)$ by $1$ .

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' (1 + \sin (x) + \sin (x) \cdot 1 + \sin (x) \sin (x))$

Step 5.3.2.4.1.3

Multiply $\sin (x)$ by $1$ .

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' (1 + \sin (x) + \sin (x) + \sin (x) \sin (x))$

Step 5.3.2.4.1.4

Multiply $\sin (x) \sin (x)$ .

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Step 5.3.2.4.1.4.1

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

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' (1 + \sin (x) + \sin (x) + \sin^{1} (x) \sin (x))$

Step 5.3.2.4.1.4.2

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

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' (1 + \sin (x) + \sin (x) + \sin^{1} (x) \sin^{1} (x))$

Step 5.3.2.4.1.4.3

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

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' (1 + \sin (x) + \sin (x) + {\sin (x)}^{1 + 1})$

Step 5.3.2.4.1.4.4

Add $1$ and $1$ .

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' (1 + \sin (x) + \sin (x) + \sin^{2} (x))$

Step 5.3.2.4.2

Add $\sin (x)$ and $\sin (x)$ .

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' (1 + 2 \sin (x) + \sin^{2} (x))$

Step 5.3.2.5

Apply the distributive property.

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' \cdot 1 + y' (2 \sin (x)) + y' (\sin^{2} (x))$

Step 5.3.2.6

Simplify.

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Step 5.3.2.6.1

Multiply $y'$ by $1$ .

$y' (e^{y}) + y' (e^{y}) \sin (x) - e^{y} \cos (x) = y' + y' (2 \sin (x)) + y' (\sin^{2} (x))$

Step 5.3.2.6.2

Rewrite using the commutative property of multiplication.

$y' e^{y} + y' e^{y} \sin (x) - e^{y} \cos (x) = y' + 2 y' \sin (x) + y' \sin^{2} (x)$

Step 5.3.3

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$y' + 2 y' \sin (x) + y' \sin^{2} (x) = y' e^{y} + y' e^{y} \sin (x) - e^{y} \cos (x)$

Step 5.3.4

Move all terms containing $y'$ to the left side of the equation.

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Step 5.3.4.1

Subtract $y' e^{y}$ from both sides of the equation.

$y' + 2 y' \sin (x) + y' \sin^{2} (x) - y' e^{y} = y' e^{y} \sin (x) - e^{y} \cos (x)$

Step 5.3.4.2

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

$y' + 2 y' \sin (x) + y' \sin^{2} (x) - y' e^{y} - y' e^{y} \sin (x) = - e^{y} \cos (x)$

Step 5.3.5

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

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Step 5.3.5.1

Factor $y'$ out of ${y'}^{1}$ .

$y' \cdot 1 + 2 y' \sin (x) + y' \sin^{2} (x) - y' e^{y} - y' e^{y} \sin (x) = - e^{y} \cos (x)$

Step 5.3.5.2

Factor $y'$ out of $2 y' \sin (x)$ .

$y' \cdot 1 + y' (2 \sin (x)) + y' (\sin^{2} (x)) - y' e^{y} - y' e^{y} \sin (x) = - e^{y} \cos (x)$

Step 5.3.5.3

Factor $y'$ out of $y' \sin^{2} (x)$ .

$y' \cdot 1 + y' (2 \sin (x)) + y' (\sin^{2} (x)) - y' e^{y} - y' e^{y} \sin (x) = - e^{y} \cos (x)$

Step 5.3.5.4

Factor $y'$ out of $- y' e^{y}$ .

$y' \cdot 1 + y' (2 \sin (x)) + y' (\sin^{2} (x)) + y' (- 1 e^{y}) - y' e^{y} \sin (x) = - e^{y} \cos (x)$

Step 5.3.5.5

Factor $y'$ out of $- y' e^{y} \sin (x)$ .

$y' \cdot 1 + y' (2 \sin (x)) + y' (\sin^{2} (x)) + y' (- 1 e^{y}) + y' (- 1 e^{y} \sin (x)) = - e^{y} \cos (x)$

Step 5.3.5.6

Factor $y'$ out of $y' \cdot 1 + y' (2 \sin (x))$ .

$y' \cdot (1 + 2 \sin (x)) + y' (\sin^{2} (x)) + y' (- 1 e^{y}) + y' (- 1 e^{y} \sin (x)) = - e^{y} \cos (x)$

Step 5.3.5.7

Factor $y'$ out of $y' \cdot (1 + 2 \sin (x)) + y' \sin^{2} (x)$ .

$y' \cdot (1 + 2 \sin (x) + \sin^{2} (x)) + y' (- 1 e^{y}) + y' (- 1 e^{y} \sin (x)) = - e^{y} \cos (x)$

Step 5.3.5.8

Factor $y'$ out of $y' \cdot (1 + 2 \sin (x) + \sin^{2} (x)) + y' (- 1 e^{y})$ .

$y' \cdot (1 + 2 \sin (x) + \sin^{2} (x) - 1 e^{y}) + y' (- 1 e^{y} \sin (x)) = - e^{y} \cos (x)$

Step 5.3.5.9

Factor $y'$ out of $y' \cdot (1 + 2 \sin (x) + \sin^{2} (x) - 1 e^{y}) + y' (- 1 e^{y} \sin (x))$ .

$y' (1 + 2 \sin (x) + \sin^{2} (x) - 1 e^{y} - 1 e^{y} \sin (x)) = - e^{y} \cos (x)$

Step 5.3.6

Rewrite $- 1 e^{y}$ as $- e^{y}$ .

$y' (1 + 2 \sin (x) + \sin^{2} (x) - e^{y} - 1 e^{y} \sin (x)) = - e^{y} \cos (x)$

Step 5.3.7

Rewrite $- 1 e^{y}$ as $- e^{y}$ .

$y' (1 + 2 \sin (x) + \sin^{2} (x) - e^{y} - e^{y} \sin (x)) = - e^{y} \cos (x)$

Step 5.3.8

Divide each term in $y' (1 + 2 \sin (x) + \sin^{2} (x) - e^{y} - e^{y} \sin (x)) = - e^{y} \cos (x)$ by $1 + 2 \sin (x) + \sin^{2} (x) - e^{y} - e^{y} \sin (x)$ and simplify.

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Step 5.3.8.1

Divide each term in $y' (1 + 2 \sin (x) + \sin^{2} (x) - e^{y} - e^{y} \sin (x)) = - e^{y} \cos (x)$ by $1 + 2 \sin (x) + \sin^{2} (x) - e^{y} - e^{y} \sin (x)$ .

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

Step 5.3.8.2

Simplify the left side.

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Step 5.3.8.2.1

Cancel the common factor of $1 + 2 \sin (x) + \sin^{2} (x) - e^{y} - e^{y} \sin (x)$ .

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Step 5.3.8.2.1.1

Cancel the common factor.

Step 5.3.8.2.1.2

Divide $y'$ by $1$ .

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

Step 5.3.8.3

Simplify the right side.

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Step 5.3.8.3.1

Move the negative in front of the fraction.

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

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

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