Calculus Examples

$\ln (y) + \frac{x}{y} = 2021$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (\ln (y) + \frac{x}{y}) = \frac{d}{d x} (2021)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.2

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

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

Step 2.2.3

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

$\frac{y'}{y} + \frac{d}{d x} [\frac{x}{y}]$

Step 2.3

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

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Step 2.3.1

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 $f (x) = x$ and $g (x) = y$ .

$\frac{y'}{y} + \frac{y \frac{d}{d x} [x] - x \frac{d}{d x} [y]}{y^{2}}$

Step 2.3.2

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

$\frac{y'}{y} + \frac{y \cdot 1 - x \frac{d}{d x} [y]}{y^{2}}$

Step 2.3.3

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

$\frac{y'}{y} + \frac{y \cdot 1 - x y'}{y^{2}}$

Step 2.3.4

Multiply $y$ by $1$ .

$\frac{y'}{y} + \frac{y - x y'}{y^{2}}$

Step 2.4

Simplify.

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Step 2.4.1

Combine terms.

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Step 2.4.1.1

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

$\frac{y'}{y} \cdot \frac{y}{y} + \frac{y - x y'}{y^{2}}$

Step 2.4.1.2

Write each expression with a common denominator of $y^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1.2.1

Multiply $\frac{y'}{y}$ by $\frac{y}{y}$ .

$\frac{y' y}{y \cdot y} + \frac{y - x y'}{y^{2}}$

Step 2.4.1.2.2

Multiply $y$ by $y$ .

$\frac{y' y}{y^{2}} + \frac{y - x y'}{y^{2}}$

Step 2.4.1.3

Combine the numerators over the common denominator.

$\frac{y' y + y - x y'}{y^{2}}$

Step 2.4.2

Reorder terms.

$\frac{y y' - x y' + y}{y^{2}}$

Step 3

Since $2021$ is constant with respect to $x$ , the derivative of $2021$ with respect to $x$ is $0$ .

$0$

Step 4

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

$\frac{y y' - x y' + y}{y^{2}} = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Set the numerator equal to zero.

$y y' - x y' + y = 0$

Step 5.2

Solve the equation for $y'$ .

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Step 5.2.1

Subtract $y$ from both sides of the equation.

$y y' - x y' = - y$

Step 5.2.2

Factor $y'$ out of $y y' - x y'$ .

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Step 5.2.2.1

Factor $y'$ out of $y y'$ .

$y' y - x y' = - y$

Step 5.2.2.2

Factor $y'$ out of $- x y'$ .

$y' (y) + y' (- x) = - y$

Step 5.2.2.3

Factor $y'$ out of $y' (y) + y' (- x)$ .

$y' (y - x) = - y$

Step 5.2.3

Divide each term in $y' (y - x) = - y$ by $y - x$ and simplify.

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Step 5.2.3.1

Divide each term in $y' (y - x) = - y$ by $y - x$ .

$\frac{y' (y - x)}{y - x} = \frac{- y}{y - x}$

Step 5.2.3.2

Simplify the left side.

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Step 5.2.3.2.1

Cancel the common factor of $y - x$ .

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Step 5.2.3.2.1.1

Cancel the common factor.

$\frac{y' (y - x)}{y - x} = \frac{- y}{y - x}$

Step 5.2.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- y}{y - x}$

Step 5.2.3.3

Simplify the right side.

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Step 5.2.3.3.1

Move the negative in front of the fraction.

$y' = - \frac{y}{y - x}$

Step 6

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

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