Calculus Examples

$y^{3} + y x^{2} + x^{2} - 3 y^{2} = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{3} + y x^{2} + x^{2} - 3 y^{2}) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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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 $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.2.1.1

To apply the Chain Rule, set $u_{1}$ as $y$ .

$\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d x} [y] + \frac{d}{d x} [y x^{2}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 y^{2}]$

Step 2.2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$3 {u_{1}}^{2} \frac{d}{d x} [y] + \frac{d}{d x} [y x^{2}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 3 y^{2}]$

Step 2.2.1.3

Replace all occurrences of $u_{1}$ with $y$ .

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.3.4

Move $2$ to the left of $y$ .

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

Step 2.4

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

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

Step 2.5

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

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Step 2.5.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 y^{2}$ with respect to $x$ is $- 3 \frac{d}{d x} [y^{2}]$ .

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

Step 2.5.2

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 2.5.2.1

To apply the Chain Rule, set $u_{2}$ as $y$ .

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

Step 2.5.2.2

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

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

Step 2.5.2.3

Replace all occurrences of $u_{2}$ with $y$ .

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

Step 2.5.3

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

$3 y^{2} y' + 2 y x + x^{2} y' + 2 x - 3 (2 y y')$

Step 2.5.4

Multiply $2$ by $- 3$ .

$3 y^{2} y' + 2 y x + x^{2} y' + 2 x - 6 y y'$

Step 2.6

Reorder terms.

$3 y^{2} y' + 2 x y + x^{2} y' - 6 y y' + 2 x$

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.

$3 y^{2} y' + 2 x y + x^{2} y' - 6 y y' + 2 x = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.1.1

Subtract $2 x y$ from both sides of the equation.

$3 y^{2} y' + x^{2} y' - 6 y y' + 2 x = - 2 x y$

Step 5.1.2

Subtract $2 x$ from both sides of the equation.

$3 y^{2} y' + x^{2} y' - 6 y y' = - 2 x y - 2 x$

Step 5.2

Factor $y'$ out of $3 y^{2} y' + x^{2} y' - 6 y y'$ .

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Step 5.2.1

Factor $y'$ out of $3 y^{2} y'$ .

$y' (3 y^{2}) + x^{2} y' - 6 y y' = - 2 x y - 2 x$

Step 5.2.2

Factor $y'$ out of $x^{2} y'$ .

$y' (3 y^{2}) + y' (x^{2}) - 6 y y' = - 2 x y - 2 x$

Step 5.2.3

Factor $y'$ out of $- 6 y y'$ .

$y' (3 y^{2}) + y' (x^{2}) + y' (- 6 y) = - 2 x y - 2 x$

Step 5.2.4

Factor $y'$ out of $y' (3 y^{2}) + y' (x^{2})$ .

$y' (3 y^{2} + x^{2}) + y' (- 6 y) = - 2 x y - 2 x$

Step 5.2.5

Factor $y'$ out of $y' (3 y^{2} + x^{2}) + y' (- 6 y)$ .

$y' (3 y^{2} + x^{2} - 6 y) = - 2 x y - 2 x$

Step 5.3

Divide each term in $y' (3 y^{2} + x^{2} - 6 y) = - 2 x y - 2 x$ by $3 y^{2} + x^{2} - 6 y$ and simplify.

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Step 5.3.1

Divide each term in $y' (3 y^{2} + x^{2} - 6 y) = - 2 x y - 2 x$ by $3 y^{2} + x^{2} - 6 y$ .

$\frac{y' (3 y^{2} + x^{2} - 6 y)}{3 y^{2} + x^{2} - 6 y} = \frac{- 2 x y}{3 y^{2} + x^{2} - 6 y} + \frac{- 2 x}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $3 y^{2} + x^{2} - 6 y$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{y' (3 y^{2} + x^{2} - 6 y)}{3 y^{2} + x^{2} - 6 y} = \frac{- 2 x y}{3 y^{2} + x^{2} - 6 y} + \frac{- 2 x}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 2 x y}{3 y^{2} + x^{2} - 6 y} + \frac{- 2 x}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Combine the numerators over the common denominator.

$y' = \frac{- 2 x y - 2 x}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.3.2

Simplify the numerator.

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Step 5.3.3.2.1

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

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Step 5.3.3.2.1.1

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

$y' = \frac{2 x (- 1 y) - 2 x}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.3.2.1.2

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

$y' = \frac{2 x (- 1 y) + 2 x (- 1)}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.3.2.1.3

Factor $2 x$ out of $2 x (- 1 y) + 2 x (- 1)$ .

$y' = \frac{2 x (- 1 y - 1)}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.3.2.2

Rewrite $- 1 y$ as $- y$ .

$y' = \frac{2 x (- y - 1)}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.3.3

Simplify with factoring out.

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Step 5.3.3.3.1

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

$y' = \frac{2 x (- (y) - 1)}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.3.3.2

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

$y' = \frac{2 x (- (y) - 1 (1))}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.3.3.3

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

$y' = \frac{2 x (- (y + 1))}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.3.3.4

Simplify the expression.

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Step 5.3.3.3.4.1

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

$y' = \frac{2 x (- 1 (y + 1))}{3 y^{2} + x^{2} - 6 y}$

Step 5.3.3.3.4.2

Move the negative in front of the fraction.

$y' = - \frac{(2 x) (y + 1)}{3 y^{2} + x^{2} - 6 y}$

Step 6

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

$\frac{d y}{d x} = - \frac{(2 x) (y + 1)}{3 y^{2} + x^{2} - 6 y}$