Calculus Examples

$\ln (y) = e^{y} \sin (x)$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

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

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [y]$

Step 2.1.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$\frac{1}{u} \frac{d}{d x} [y]$

Step 2.1.3

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

$\frac{1}{y} \frac{d}{d x} [y]$

Step 2.2

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

$\frac{1}{y} y'$

Step 2.3

Combine $\frac{1}{y}$ and $y'$ .

$\frac{y'}{y}$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

Reorder terms.

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by $y$ .

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

Step 5.2

Simplify.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

Cancel the common factor of $y$ .

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Step 5.2.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.2

Rewrite the expression.

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

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

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

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Step 5.2.2.1.1

Apply the distributive property.

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

Step 5.2.2.1.2

Simplify the expression.

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Step 5.2.2.1.2.1

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

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

Step 5.2.2.1.2.2

Reorder $y e^{y} \cos (x)$ and $y' y e^{y} \sin (x)$ .

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

Step 5.3

Solve for $y'$ .

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Step 5.3.1

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

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

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

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

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

Step 5.3.2.3

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

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

Step 5.3.3

Rewrite $- 1 y$ as $- y$ .

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

Step 5.3.4

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

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Step 5.3.4.1

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

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

Step 5.3.4.2

Simplify the left side.

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Step 5.3.4.2.1

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

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Step 5.3.4.2.1.1

Cancel the common factor.

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

Step 5.3.4.2.1.2

Divide $y'$ by $1$ .

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

Step 6

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

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