Algebra Examples

$x^{5} = - x y^{3}$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$5 x^{4}$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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 $f (x) = x$ and $g (x) = y^{3}$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.4

Move $3$ to the left of $x$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

Multiply $y^{3}$ by $1$ .

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

Step 3.8

Simplify.

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Step 3.8.1

Apply the distributive property.

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

Step 3.8.2

Multiply $3$ by $- 1$ .

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

Step 3.8.3

Reorder terms.

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

Step 4

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

$5 x^{4} = - y^{3} - 3 y^{2} x y'$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $- y^{3} - 3 y^{2} x y' = 5 x^{4}$ .

$- y^{3} - 3 y^{2} x y' = 5 x^{4}$

Step 5.2

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

$- 3 y^{2} x y' = 5 x^{4} + y^{3}$

Step 5.3

Divide each term in $- 3 y^{2} x y' = 5 x^{4} + y^{3}$ by $- 3 y^{2} x$ and simplify.

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Step 5.3.1

Divide each term in $- 3 y^{2} x y' = 5 x^{4} + y^{3}$ by $- 3 y^{2} x$ .

$\frac{- 3 y^{2} x y'}{- 3 y^{2} x} = \frac{5 x^{4}}{- 3 y^{2} x} + \frac{y^{3}}{- 3 y^{2} x}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 3$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 3 y^{2} x y'}{- 3 y^{2} x} = \frac{5 x^{4}}{- 3 y^{2} x} + \frac{y^{3}}{- 3 y^{2} x}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y^{2} x y'}{y^{2} x} = \frac{5 x^{4}}{- 3 y^{2} x} + \frac{y^{3}}{- 3 y^{2} x}$

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.

$\frac{y^{2} x y'}{y^{2} x} = \frac{5 x^{4}}{- 3 y^{2} x} + \frac{y^{3}}{- 3 y^{2} x}$

Step 5.3.2.2.2

Rewrite the expression.

$\frac{x y'}{x} = \frac{5 x^{4}}{- 3 y^{2} x} + \frac{y^{3}}{- 3 y^{2} x}$

Step 5.3.2.3

Cancel the common factor of $x$ .

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Step 5.3.2.3.1

Cancel the common factor.

$\frac{x y'}{x} = \frac{5 x^{4}}{- 3 y^{2} x} + \frac{y^{3}}{- 3 y^{2} x}$

Step 5.3.2.3.2

Divide $y'$ by $1$ .

$y' = \frac{5 x^{4}}{- 3 y^{2} x} + \frac{y^{3}}{- 3 y^{2} x}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $x^{4}$ and $x$ .

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Step 5.3.3.1.1.1

Factor $x$ out of $5 x^{4}$ .

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

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

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

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

Step 5.3.3.1.1.2.2

Cancel the common factor.

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

Step 5.3.3.1.1.2.3

Rewrite the expression.

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

Step 5.3.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.3.1.3

Cancel the common factor of $y^{3}$ and $y^{2}$ .

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Step 5.3.3.1.3.1

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

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

Step 5.3.3.1.3.2

Cancel the common factors.

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Step 5.3.3.1.3.2.1

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

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

Step 5.3.3.1.3.2.2

Cancel the common factor.

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

Step 5.3.3.1.3.2.3

Rewrite the expression.

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

Step 5.3.3.1.4

Move the negative in front of the fraction.

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

Step 6

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

$\frac{d y}{d x} = - \frac{5 x^{3}}{3 y^{2}} - \frac{y}{3 x}$