Calculus Examples

$- 7 x y + 3 y - 7 = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (- 7 x y + 3 y - 7) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $- 7 x y + 3 y - 7$ with respect to $x$ is $\frac{d}{d x} [- 7 x y] + \frac{d}{d x} [3 y] + \frac{d}{d x} [- 7]$ .

$\frac{d}{d x} [- 7 x y] + \frac{d}{d x} [3 y] + \frac{d}{d x} [- 7]$

Step 2.2

Evaluate $\frac{d}{d x} [- 7 x y]$ .

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Step 2.2.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x y$ with respect to $x$ is $- 7 \frac{d}{d x} [x y]$ .

$- 7 \frac{d}{d x} [x y] + \frac{d}{d x} [3 y] + \frac{d}{d x} [- 7]$

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 $g (x) = y$ .

$- 7 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x]) + \frac{d}{d x} [3 y] + \frac{d}{d x} [- 7]$

Step 2.2.3

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

$- 7 (x y' + y \frac{d}{d x} [x]) + \frac{d}{d x} [3 y] + \frac{d}{d x} [- 7]$

Step 2.2.4

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

$- 7 (x y' + y \cdot 1) + \frac{d}{d x} [3 y] + \frac{d}{d x} [- 7]$

Step 2.2.5

Multiply $y$ by $1$ .

$- 7 (x y' + y) + \frac{d}{d x} [3 y] + \frac{d}{d x} [- 7]$

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

$- 7 (x y' + y) + 3 \frac{d}{d x} [y] + \frac{d}{d x} [- 7]$

Step 2.3.2

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

$- 7 (x y' + y) + 3 y' + \frac{d}{d x} [- 7]$

Step 2.4

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$- 7 (x y' + y) + 3 y' + 0$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$- 7 (x y') - 7 y + 3 y' + 0$

Step 2.5.2

Add $- 7 x y' - 7 y + 3 y'$ and $0$ .

$- 7 x y' - 7 y + 3 y'$

Step 3

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

$0$

Step 4

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

$- 7 x y' - 7 y + 3 y' = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Add $7 y$ to both sides of the equation.

$- 7 x y' + 3 y' = 7 y$

Step 5.2

Factor $y'$ out of $- 7 x y' + 3 y'$ .

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Step 5.2.1

Factor $y'$ out of $- 7 x y'$ .

$y' (- 7 x) + 3 y' = 7 y$

Step 5.2.2

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

$y' (- 7 x) + y' \cdot 3 = 7 y$

Step 5.2.3

Factor $y'$ out of $y' (- 7 x) + y' \cdot 3$ .

$y' (- 7 x + 3) = 7 y$

Step 5.3

Divide each term in $y' (- 7 x + 3) = 7 y$ by $- 7 x + 3$ and simplify.

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Step 5.3.1

Divide each term in $y' (- 7 x + 3) = 7 y$ by $- 7 x + 3$ .

$\frac{y' (- 7 x + 3)}{- 7 x + 3} = \frac{7 y}{- 7 x + 3}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 7 x + 3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{y' (- 7 x + 3)}{- 7 x + 3} = \frac{7 y}{- 7 x + 3}$

Step 5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{7 y}{- 7 x + 3}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Factor $- 1$ out of $- 7 x$ .

$y' = \frac{7 y}{- (7 x) + 3}$

Step 5.3.3.2

Rewrite $3$ as $- 1 (- 3)$ .

$y' = \frac{7 y}{- (7 x) - 1 (- 3)}$

Step 5.3.3.3

Factor $- 1$ out of $- (7 x) - 1 (- 3)$ .

$y' = \frac{7 y}{- (7 x - 3)}$

Step 5.3.3.4

Rewrite negatives.

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Step 5.3.3.4.1

Rewrite $- (7 x - 3)$ as $- 1 (7 x - 3)$ .

$y' = \frac{7 y}{- 1 (7 x - 3)}$

Step 5.3.3.4.2

Move the negative in front of the fraction.

$y' = - \frac{7 y}{7 x - 3}$

Step 6

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

$\frac{d y}{d x} = - \frac{7 y}{7 x - 3}$