Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$0$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{3}$ with respect to $x$ is $- \frac{d}{d x} [x^{3}]$ .

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $3$ by $- 1$ .

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

Step 3.3

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

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Step 3.3.1

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 3.3.1.1

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

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

Step 3.3.1.2

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

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

Step 3.3.1.3

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

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

Step 3.3.2

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

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

Step 3.4

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

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Step 3.4.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 3.4.2

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

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

Step 3.4.3

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $- 3 x^{2} + 3 y^{2} y' - 2 x = 0$ .

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

Step 5.2

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

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Step 5.2.1

Add $3 x^{2}$ to both sides of the equation.

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

Step 5.2.2

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

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

Step 5.3

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

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Step 5.3.1

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

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

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.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of $y^{2}$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Cancel the common factor of $3$ .

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Step 5.3.3.1.1

Cancel the common factor.

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

Step 5.3.3.1.2

Rewrite the expression.

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

Step 6

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

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