Calculus Examples

$5 - 2 x y = 7 x - 3 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (5 - 2 x y) = \frac{d}{d x} (7 x - 3 y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $5 - 2 x y$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- 2 x y]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [- 2 x y]$

Step 2.1.2

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

$0 + \frac{d}{d x} [- 2 x y]$

Step 2.2

Evaluate $\frac{d}{d x} [- 2 x y]$ .

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Step 2.2.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x y$ with respect to $x$ is $- 2 \frac{d}{d x} [x y]$ .

$0 - 2 \frac{d}{d x} [x y]$

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 $g (x) = y$ .

$0 - 2 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 2.2.3

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

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

Step 2.2.4

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

$0 - 2 (x y' + y \cdot 1)$

Step 2.2.5

Multiply $y$ by $1$ .

$0 - 2 (x y' + y)$

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

$0 - 2 (x y') - 2 y$

Step 2.3.2

Subtract $- (- 2 x y' - 2 y)$ from $0$ .

$- 2 x y' - 2 y$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $7$ by $1$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

$7 - 3 y'$

Step 4

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

$- 2 x y' - 2 y = 7 - 3 y'$

Step 5

Solve for $y'$ .

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Step 5.1

Add $3 y'$ to both sides of the equation.

$- 2 x y' - 2 y + 3 y' = 7$

Step 5.2

Add $2 y$ to both sides of the equation.

$- 2 x y' + 3 y' = 7 + 2 y$

Step 5.3

Factor $y'$ out of $- 2 x y' + 3 y'$ .

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Step 5.3.1

Factor $y'$ out of $- 2 x y'$ .

$y' (- 2 x) + 3 y' = 7 + 2 y$

Step 5.3.2

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

$y' (- 2 x) + y' \cdot 3 = 7 + 2 y$

Step 5.3.3

Factor $y'$ out of $y' (- 2 x) + y' \cdot 3$ .

$y' (- 2 x + 3) = 7 + 2 y$

Step 5.4

Divide each term in $y' (- 2 x + 3) = 7 + 2 y$ by $- 2 x + 3$ and simplify.

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Step 5.4.1

Divide each term in $y' (- 2 x + 3) = 7 + 2 y$ by $- 2 x + 3$ .

$\frac{y' (- 2 x + 3)}{- 2 x + 3} = \frac{7}{- 2 x + 3} + \frac{2 y}{- 2 x + 3}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $- 2 x + 3$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{y' (- 2 x + 3)}{- 2 x + 3} = \frac{7}{- 2 x + 3} + \frac{2 y}{- 2 x + 3}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{7}{- 2 x + 3} + \frac{2 y}{- 2 x + 3}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

$y' = \frac{7 + 2 y}{- 2 x + 3}$

Step 5.4.3.2

Factor $- 1$ out of $- 2 x$ .

$y' = \frac{7 + 2 y}{- (2 x) + 3}$

Step 5.4.3.3

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

$y' = \frac{7 + 2 y}{- (2 x) - 1 (- 3)}$

Step 5.4.3.4

Factor $- 1$ out of $- (2 x) - 1 (- 3)$ .

$y' = \frac{7 + 2 y}{- (2 x - 3)}$

Step 5.4.3.5

Rewrite negatives.

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Step 5.4.3.5.1

Rewrite $- (2 x - 3)$ as $- 1 (2 x - 3)$ .

$y' = \frac{7 + 2 y}{- 1 (2 x - 3)}$

Step 5.4.3.5.2

Move the negative in front of the fraction.

$y' = - \frac{7 + 2 y}{2 x - 3}$

Step 6

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

$\frac{d y}{d x} = - \frac{7 + 2 y}{2 x - 3}$