Calculus Examples

$y^{2} - 3 x y = 2$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $y^{2} - 3 x y$ with respect to $x$ is $\frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 3 x y]$ .

$\frac{d}{d x} [y^{2}] + \frac{d}{d x} [- 3 x y]$

Step 2.2

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

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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 $f (x) = x^{2}$ and $g (x) = y$ .

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Step 2.2.1.1

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

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [y] + \frac{d}{d x} [- 3 x y]$

Step 2.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$2 u \frac{d}{d x} [y] + \frac{d}{d x} [- 3 x y]$

Step 2.2.1.3

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

$2 y \frac{d}{d x} [y] + \frac{d}{d x} [- 3 x y]$

Step 2.2.2

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

$2 y y' + \frac{d}{d x} [- 3 x y]$

Step 2.3

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

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Step 2.3.1

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

$2 y y' - 3 \frac{d}{d x} [x y]$

Step 2.3.2

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 $g (x) = y$ .

$2 y y' - 3 (x \frac{d}{d x} [y] + y \frac{d}{d x} [x])$

Step 2.3.3

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

$2 y y' - 3 (x y' + y \frac{d}{d x} [x])$

Step 2.3.4

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

$2 y y' - 3 (x y' + y \cdot 1)$

Step 2.3.5

Multiply $y$ by $1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$2 y y' - 3 (x y') - 3 y$

Step 2.4.2

Remove unnecessary parentheses.

$2 y y' - 3 x y' - 3 y$

Step 3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0$

Step 4

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

$2 y y' - 3 x y' - 3 y = 0$

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

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

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Divide $y'$ by $1$ .

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

Step 6

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

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

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 7.1

Set the numerator equal to zero.

$3 y = 0$

Step 7.2

Divide each term in $3 y = 0$ by $3$ and simplify.

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Step 7.2.1

Divide each term in $3 y = 0$ by $3$ .

$\frac{3 y}{3} = \frac{0}{3}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $3$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{0}{3}$

Step 7.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{0}{3}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $0$ by $3$ .

$y = 0$

Step 7.3

The variable $x$ got canceled.

All real numbers

Step 8

Find the points where $\frac{d y}{d x} = 0$ .

$(A l \cdot l (r e a l) (n u m b e r s), A l \cdot l (r e a l) (n u m b e r s))$

Step 9