Calculus Examples

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

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{3} - 2 x^{2}) = \frac{d}{d x} (4 - y^{2})$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

$3 x^{2} - 2 (2 x)$

Step 2.2.3

Multiply $2$ by $- 2$ .

$3 x^{2} - 4 x$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $4 - y^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- y^{2}]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- y^{2}]$

Step 3.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- y^{2}]$

Step 3.2

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

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Step 3.2.1

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

$0 - \frac{d}{d x} [y^{2}]$

Step 3.2.2

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 3.2.2.1

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

$0 - (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y])$

Step 3.2.2.2

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

$0 - (2 u \frac{d}{d x} [y])$

Step 3.2.2.3

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

$0 - (2 y \frac{d}{d x} [y])$

Step 3.2.3

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

$0 - (2 y y')$

Step 3.2.4

Multiply $2$ by $- 1$ .

$0 - 2 y y'$

Step 3.3

Subtract $2 y y'$ from $0$ .

$- 2 y y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $- 2 y y' = 3 x^{2} - 4 x$ .

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Rewrite the expression.

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

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.

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

Step 5.2.2.2.2

Divide $y'$ by $1$ .

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

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

Move the negative in front of the fraction.

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

Step 5.2.3.1.2

Cancel the common factor of $- 4$ and $- 2$ .

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Step 5.2.3.1.2.1

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

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

Step 5.2.3.1.2.2

Cancel the common factors.

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Step 5.2.3.1.2.2.1

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

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

Step 5.2.3.1.2.2.2

Cancel the common factor.

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

Step 5.2.3.1.2.2.3

Rewrite the expression.

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

Step 6

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

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