Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

Differentiate.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Add $0$ and $\frac{d}{d x} [- x]$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

Simplify the expression.

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Step 2.2.6.1

Multiply $- 1$ by $1$ .

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

Step 2.2.6.2

Move $- 1$ to the left of $y^{2}$ .

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

Step 2.2.6.3

Rewrite $- 1 y^{2}$ as $- y^{2}$ .

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

Step 2.3

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 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

Move $2$ to the left of $2 - x$ .

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

Step 2.5

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

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

Step 2.6

Simplify.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Apply the distributive property.

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

Step 2.6.3

Apply the distributive property.

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

Step 2.6.4

Combine terms.

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Step 2.6.4.1

Multiply $2$ by $2$ .

$- y^{2} + 4 y y' + 2 (- x) y y'$

Step 2.6.4.2

Multiply $- 1$ by $2$ .

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

Step 3

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

$3 x^{2}$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

Factor $2 y y'$ out of $4 y y'$ .

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

Step 5.2.2

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

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

Step 5.2.3

Factor $2 y y'$ out of $2 y y' \cdot 2 + 2 y y' (- x)$ .

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

Cancel the common factor of $y$ .

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Rewrite the expression.

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

Step 5.3.2.3

Cancel the common factor of $2 - x$ .

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Step 5.3.2.3.1

Cancel the common factor.

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

Step 5.3.2.3.2

Divide $y'$ by $1$ .

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

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

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

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Step 5.3.3.1.1

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

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

Step 5.3.3.1.2

Cancel the common factors.

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Step 5.3.3.1.2.1

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

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

Step 5.3.3.1.2.2

Cancel the common factor.

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

Step 5.3.3.1.2.3

Rewrite the expression.

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

Step 6

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

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