Calculus Examples

$3 x y + 3 y + 5 = - 3 x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 x y + 3 y + 5) = \frac{d}{d x} (- 3 x)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $3 x y + 3 y + 5$ with respect to $x$ is $\frac{d}{d x} [3 x y] + \frac{d}{d x} [3 y] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [3 x y] + \frac{d}{d x} [3 y] + \frac{d}{d x} [5]$

Step 2.2

Evaluate $\frac{d}{d x} [3 x y]$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x y$ with respect to $x$ is $3 \frac{d}{d x} [x y]$ .

$3 \frac{d}{d x} [x y] + \frac{d}{d x} [3 y] + \frac{d}{d x} [5]$

Step 2.2.2

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

$3 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [3 y] + \frac{d}{d x} [5]$

Step 2.2.3

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

$3 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [3 y] + \frac{d}{d x} [5]$

Step 2.2.4

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

$3 (x y' + y \cdot 1) + \frac{d}{d x} [3 y] + \frac{d}{d x} [5]$

Step 2.2.5

Multiply $y$ by $1$ .

$3 (x y' + y) + \frac{d}{d x} [3 y] + \frac{d}{d x} [5]$

Step 2.3

Evaluate $\frac{d}{d x} [3 y]$ .

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Step 2.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 y$ with respect to $x$ is $3 \frac{d}{d x} [y]$ .

$3 (x y' + y) + 3 \frac{d}{d x} [y] + \frac{d}{d x} [5]$

Step 2.3.2

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

$3 (x y' + y) + 3 y' + \frac{d}{d x} [5]$

Step 2.4

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$3 (x y' + y) + 3 y' + 0$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$3 (x y') + 3 y + 3 y' + 0$

Step 2.5.2

Add $3 x y' + 3 y + 3 y'$ and $0$ .

$3 x y' + 3 y + 3 y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$- 3 \frac{d}{d x} [x]$

Step 3.2

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

$- 3 \cdot 1$

Step 3.3

Multiply $- 3$ by $1$ .

$- 3$

Step 4

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

$3 x y' + 3 y + 3 y' = - 3$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $3 y$ from both sides of the equation.

$3 x y' + 3 y' = - 3 - 3 y$

Step 5.2

Factor $3 y'$ out of $3 x y' + 3 y'$ .

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Step 5.2.1

Factor $3 y'$ out of $3 x y'$ .

$3 y' x + 3 y' = - 3 - 3 y$

Step 5.2.2

Factor $3 y'$ out of $3 y'$ .

$3 y' x + 3 y' \cdot 1 = - 3 - 3 y$

Step 5.2.3

Factor $3 y'$ out of $3 y' x + 3 y' \cdot 1$ .

$3 y' (x + 1) = - 3 - 3 y$

Step 5.3

Divide each term in $3 y' (x + 1) = - 3 - 3 y$ by $3 (x + 1)$ and simplify.

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Step 5.3.1

Divide each term in $3 y' (x + 1) = - 3 - 3 y$ by $3 (x + 1)$ .

$\frac{3 y' (x + 1)}{3 (x + 1)} = \frac{- 3}{3 (x + 1)} + \frac{- 3 y}{3 (x + 1)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{3 y' (x + 1)}{3 (x + 1)} = \frac{- 3}{3 (x + 1)} + \frac{- 3 y}{3 (x + 1)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' (x + 1)}{x + 1} = \frac{- 3}{3 (x + 1)} + \frac{- 3 y}{3 (x + 1)}$

Step 5.3.2.2

Cancel the common factor of $x + 1$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' (x + 1)}{x + 1} = \frac{- 3}{3 (x + 1)} + \frac{- 3 y}{3 (x + 1)}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 3}{3 (x + 1)} + \frac{- 3 y}{3 (x + 1)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $- 3$ and $3$ .

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Step 5.3.3.1.1.1

Factor $3$ out of $- 3$ .

$y' = \frac{3 \cdot - 1}{3 (x + 1)} + \frac{- 3 y}{3 (x + 1)}$

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Cancel the common factor.

$y' = \frac{3 \cdot - 1}{3 (x + 1)} + \frac{- 3 y}{3 (x + 1)}$

Step 5.3.3.1.1.2.2

Rewrite the expression.

$y' = \frac{- 1}{x + 1} + \frac{- 3 y}{3 (x + 1)}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{1}{x + 1} + \frac{- 3 y}{3 (x + 1)}$

Step 5.3.3.1.3

Cancel the common factor of $- 3$ and $3$ .

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Step 5.3.3.1.3.1

Factor $3$ out of $- 3 y$ .

$y' = - \frac{1}{x + 1} + \frac{3 (- y)}{3 (x + 1)}$

Step 5.3.3.1.3.2

Cancel the common factors.

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Step 5.3.3.1.3.2.1

Cancel the common factor.

$y' = - \frac{1}{x + 1} + \frac{3 (- y)}{3 (x + 1)}$

Step 5.3.3.1.3.2.2

Rewrite the expression.

$y' = - \frac{1}{x + 1} + \frac{- y}{x + 1}$

Step 5.3.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{1}{x + 1} - \frac{y}{x + 1}$

Step 5.3.3.2

Simplify terms.

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Step 5.3.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{- 1 - y}{x + 1}$

Step 5.3.3.2.2

Rewrite $- 1$ as $- 1 (1)$ .

$y' = \frac{- 1 (1) - y}{x + 1}$

Step 5.3.3.2.3

Factor $- 1$ out of $- y$ .

$y' = \frac{- 1 (1) - (y)}{x + 1}$

Step 5.3.3.2.4

Factor $- 1$ out of $- 1 (1) - (y)$ .

$y' = \frac{- 1 (1 + y)}{x + 1}$

Step 5.3.3.2.5

Move the negative in front of the fraction.

$y' = - \frac{1 + y}{x + 1}$

Step 6

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

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