Calculus Examples

$- x^{3} + 5 x - 3 y^{2} + 2 = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (- x^{3} + 5 x - 3 y^{2} + 2) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $- x^{3} + 5 x - 3 y^{2} + 2$ with respect to $x$ is $\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 3 y^{2}] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [- x^{3}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 3 y^{2}] + \frac{d}{d x} [2]$

Step 2.2

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

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Step 2.2.1

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

$- \frac{d}{d x} [x^{3}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 3 y^{2}] + \frac{d}{d x} [2]$

Step 2.2.2

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

$- (3 x^{2}) + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 3 y^{2}] + \frac{d}{d x} [2]$

Step 2.2.3

Multiply $3$ by $- 1$ .

$- 3 x^{2} + \frac{d}{d x} [5 x] + \frac{d}{d x} [- 3 y^{2}] + \frac{d}{d x} [2]$

Step 2.3

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 2.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$- 3 x^{2} + 5 \frac{d}{d x} [x] + \frac{d}{d x} [- 3 y^{2}] + \frac{d}{d x} [2]$

Step 2.3.2

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

$- 3 x^{2} + 5 \cdot 1 + \frac{d}{d x} [- 3 y^{2}] + \frac{d}{d x} [2]$

Step 2.3.3

Multiply $5$ by $1$ .

$- 3 x^{2} + 5 + \frac{d}{d x} [- 3 y^{2}] + \frac{d}{d x} [2]$

Step 2.4

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

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Step 2.4.1

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

$- 3 x^{2} + 5 - 3 \frac{d}{d x} [y^{2}] + \frac{d}{d x} [2]$

Step 2.4.2

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 2.4.2.1

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

$- 3 x^{2} + 5 - 3 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [2]$

Step 2.4.2.2

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

$- 3 x^{2} + 5 - 3 (2 u \frac{d}{d x} [y]) + \frac{d}{d x} [2]$

Step 2.4.2.3

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

$- 3 x^{2} + 5 - 3 (2 y \frac{d}{d x} [y]) + \frac{d}{d x} [2]$

Step 2.4.3

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

$- 3 x^{2} + 5 - 3 (2 y y') + \frac{d}{d x} [2]$

Step 2.4.4

Multiply $2$ by $- 3$ .

$- 3 x^{2} + 5 - 6 y y' + \frac{d}{d x} [2]$

Step 2.5

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

$- 3 x^{2} + 5 - 6 y y' + 0$

Step 2.6

Simplify.

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Step 2.6.1

Add $- 3 x^{2} + 5 - 6 y y'$ and $0$ .

$- 3 x^{2} + 5 - 6 y y'$

Step 2.6.2

Reorder terms.

$- 3 x^{2} - 6 y y' + 5$

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.

$- 3 x^{2} - 6 y y' + 5 = 0$

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

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

$- 6 y y' + 5 = 3 x^{2}$

Step 5.1.2

Subtract $5$ from both sides of the equation.

$- 6 y y' = 3 x^{2} - 5$

Step 5.2

Divide each term in $- 6 y y' = 3 x^{2} - 5$ by $- 6 y$ and simplify.

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Step 5.2.1

Divide each term in $- 6 y y' = 3 x^{2} - 5$ by $- 6 y$ .

$\frac{- 6 y y'}{- 6 y} = \frac{3 x^{2}}{- 6 y} + \frac{- 5}{- 6 y}$

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.

$\frac{- 6 y y'}{- 6 y} = \frac{3 x^{2}}{- 6 y} + \frac{- 5}{- 6 y}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{y y'}{y} = \frac{3 x^{2}}{- 6 y} + \frac{- 5}{- 6 y}$

Step 5.2.2.2

Cancel the common factor of $y$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{y y'}{y} = \frac{3 x^{2}}{- 6 y} + \frac{- 5}{- 6 y}$

Step 5.2.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{3 x^{2}}{- 6 y} + \frac{- 5}{- 6 y}$

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 $3$ and $- 6$ .

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Step 5.2.3.1.1.1

Factor $3$ out of $3 x^{2}$ .

$y' = \frac{3 (x^{2})}{- 6 y} + \frac{- 5}{- 6 y}$

Step 5.2.3.1.1.2

Cancel the common factors.

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Step 5.2.3.1.1.2.1

Factor $3$ out of $- 6 y$ .

$y' = \frac{3 (x^{2})}{3 (- 2 y)} + \frac{- 5}{- 6 y}$

Step 5.2.3.1.1.2.2

Cancel the common factor.

$y' = \frac{3 x^{2}}{3 (- 2 y)} + \frac{- 5}{- 6 y}$

Step 5.2.3.1.1.2.3

Rewrite the expression.

$y' = \frac{x^{2}}{- 2 y} + \frac{- 5}{- 6 y}$

Step 5.2.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{x^{2}}{2 y} + \frac{- 5}{- 6 y}$

Step 5.2.3.1.3

Dividing two negative values results in a positive value.

$y' = - \frac{x^{2}}{2 y} + \frac{5}{6 y}$

Step 6

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

$\frac{d y}{d x} = - \frac{x^{2}}{2 y} + \frac{5}{6 y}$