Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $3$ by $1$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = y$ .

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Step 2.3.2.1

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

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

Step 2.3.2.2

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

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

Step 2.3.2.3

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

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

Step 2.3.3

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

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

Step 2.3.4

Multiply $3$ by $- 2$ .

$3 - 6 y^{2} y' + \frac{d}{d x} [2 x^{2}]$

Step 2.4

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

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Step 2.4.1

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

$3 - 6 y^{2} y' + 2 \frac{d}{d x} [x^{2}]$

Step 2.4.2

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

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

Step 2.4.3

Multiply $2$ by $2$ .

$3 - 6 y^{2} y' + 4 x$

Step 2.5

Reorder terms.

$- 6 y^{2} y' + 4 x + 3$

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.

$- 6 y^{2} y' + 4 x + 3 = 0$

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 $4 x$ from both sides of the equation.

$- 6 y^{2} y' + 3 = - 4 x$

Step 5.1.2

Subtract $3$ from both sides of the equation.

$- 6 y^{2} y' = - 4 x - 3$

Step 5.2

Divide each term in $- 6 y^{2} y' = - 4 x - 3$ by $- 6 y^{2}$ and simplify.

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Step 5.2.1

Divide each term in $- 6 y^{2} y' = - 4 x - 3$ by $- 6 y^{2}$ .

$\frac{- 6 y^{2} y'}{- 6 y^{2}} = \frac{- 4 x}{- 6 y^{2}} + \frac{- 3}{- 6 y^{2}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 6$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 6 y^{2} y'}{- 6 y^{2}} = \frac{- 4 x}{- 6 y^{2}} + \frac{- 3}{- 6 y^{2}}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{y^{2} y'}{y^{2}} = \frac{- 4 x}{- 6 y^{2}} + \frac{- 3}{- 6 y^{2}}$

Step 5.2.2.2

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

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{y^{2} y'}{y^{2}} = \frac{- 4 x}{- 6 y^{2}} + \frac{- 3}{- 6 y^{2}}$

Step 5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 4 x}{- 6 y^{2}} + \frac{- 3}{- 6 y^{2}}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Cancel the common factor of $- 4$ and $- 6$ .

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Step 5.2.3.1.1.1

Factor $- 2$ out of $- 4 x$ .

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

Step 5.2.3.1.1.2

Cancel the common factors.

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Step 5.2.3.1.1.2.1

Factor $- 2$ out of $- 6 y^{2}$ .

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

Step 5.2.3.1.1.2.2

Cancel the common factor.

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

Step 5.2.3.1.1.2.3

Rewrite the expression.

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

Step 5.2.3.1.2

Cancel the common factor of $- 3$ and $- 6$ .

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Step 5.2.3.1.2.1

Factor $- 3$ out of $- 3$ .

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

Step 5.2.3.1.2.2

Cancel the common factors.

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Step 5.2.3.1.2.2.1

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

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

Step 5.2.3.1.2.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.2.3

Rewrite the expression.

$y' = \frac{2 x}{3 y^{2}} + \frac{1}{2 y^{2}}$

Step 6

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

$\frac{d y}{d x} = \frac{2 x}{3 y^{2}} + \frac{1}{2 y^{2}}$