Calculus Examples

$a x^{5} + x^{3} y - y^{2} x = 12$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (a x^{5} + x^{3} y - y^{2} x) = \frac{d}{d x} (12)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

$a (5 x^{4}) + \frac{d}{d x} [x^{3} y] + \frac{d}{d x} [- y^{2} x]$

Step 2.2.3

Move $5$ to the left of $a$ .

$5 a x^{4} + \frac{d}{d x} [x^{3} y] + \frac{d}{d x} [- y^{2} x]$

Step 2.3

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

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Step 2.3.1

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

$5 a x^{4} + x^{3} \frac{d}{d x} [y] + y \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- y^{2} x]$

Step 2.3.2

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

$5 a x^{4} + x^{3} y' + y \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- y^{2} x]$

Step 2.3.3

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

$5 a x^{4} + x^{3} y' + y (3 x^{2}) + \frac{d}{d x} [- y^{2} x]$

Step 2.3.4

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

$5 a x^{4} + x^{3} y' + 3 y x^{2} + \frac{d}{d x} [- y^{2} x]$

Step 2.4

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

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Step 2.4.1

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

$5 a x^{4} + x^{3} y' + 3 y x^{2} - \frac{d}{d x} [y^{2} x]$

Step 2.4.2

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

$5 a x^{4} + x^{3} y' + 3 y x^{2} - (y^{2} \frac{d}{d x} [x] + x \frac{d}{d x} [y^{2}])$

Step 2.4.3

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

$5 a x^{4} + x^{3} y' + 3 y x^{2} - (y^{2} \cdot 1 + x \frac{d}{d x} [y^{2}])$

Step 2.4.4

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 2.4.4.1

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

$5 a x^{4} + x^{3} y' + 3 y x^{2} - (y^{2} \cdot 1 + x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]))$

Step 2.4.4.2

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

$5 a x^{4} + x^{3} y' + 3 y x^{2} - (y^{2} \cdot 1 + x (2 u \frac{d}{d x} [y]))$

Step 2.4.4.3

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

$5 a x^{4} + x^{3} y' + 3 y x^{2} - (y^{2} \cdot 1 + x (2 y \frac{d}{d x} [y]))$

Step 2.4.5

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

$5 a x^{4} + x^{3} y' + 3 y x^{2} - (y^{2} \cdot 1 + x (2 y y'))$

Step 2.4.6

Multiply $y^{2}$ by $1$ .

$5 a x^{4} + x^{3} y' + 3 y x^{2} - (y^{2} + x (2 y y'))$

Step 2.4.7

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

$5 a x^{4} + x^{3} y' + 3 y x^{2} - (y^{2} + 2 x y y')$

Step 2.5

Simplify.

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Step 2.5.1

Apply the distributive property.

$5 a x^{4} + x^{3} y' + 3 y x^{2} - y^{2} - (2 x y y')$

Step 2.5.2

Multiply $2$ by $- 1$ .

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

Step 2.5.3

Reorder terms.

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

Step 3

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$0$

Step 4

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

$- y^{2} + 5 x^{4} a + x^{3} y' + 3 x^{2} y - 2 x y y' = 0$

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

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

Step 5.1.2

Subtract $5 x^{4} a$ from both sides of the equation.

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

Step 5.1.3

Subtract $3 x^{2} y$ from both sides of the equation.

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

Step 5.2

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

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Step 5.2.1

Factor $x y'$ out of $x^{3} y'$ .

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

Step 5.2.2

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

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

Step 5.2.3

Factor $x y'$ out of $x y' x^{2} + x y' (- 2 y)$ .

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $x$ .

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Step 5.3.2.1.1

Cancel the common factor.

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

Step 5.3.2.1.2

Rewrite the expression.

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

Step 5.3.2.2

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

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Step 5.3.2.2.1

Cancel the common factor.

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

Step 5.3.2.2.2

Divide $y'$ by $1$ .

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

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 $- 5 x^{4} a$ .

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

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Cancel the common factor.

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

Step 5.3.3.1.1.2.2

Rewrite the expression.

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

Step 5.3.3.1.2

Move the negative in front of the fraction.

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

Step 5.3.3.1.3

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

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Step 5.3.3.1.3.1

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

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

Step 5.3.3.1.3.2

Cancel the common factors.

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Step 5.3.3.1.3.2.1

Cancel the common factor.

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

Step 5.3.3.1.3.2.2

Rewrite the expression.

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

Step 5.3.3.1.4

Move the negative in front of the fraction.

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

Step 5.3.3.2

To write $- \frac{5 x^{3} a}{x^{2} - 2 y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.3.3.3

Write each expression with a common denominator of $x (x^{2} - 2 y)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.3.1

Multiply $\frac{5 x^{3} a}{x^{2} - 2 y}$ by $\frac{x}{x}$ .

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

Step 5.3.3.3.2

Reorder the factors of $(x^{2} - 2 y) x$ .

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

Step 5.3.3.4

Combine the numerators over the common denominator.

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

Step 5.3.3.5

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 5.3.3.5.1

Move $x$ .

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

Step 5.3.3.5.2

Multiply $x$ by $x^{3}$ .

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Step 5.3.3.5.2.1

Raise $x$ to the power of $1$ .

$y' = \frac{y^{2} - 5 (x^{1} x^{3}) a}{x (x^{2} - 2 y)} - \frac{3 x y}{x^{2} - 2 y}$

Step 5.3.3.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y' = \frac{y^{2} - 5 x^{1 + 3} a}{x (x^{2} - 2 y)} - \frac{3 x y}{x^{2} - 2 y}$

Step 5.3.3.5.3

Add $1$ and $3$ .

$y' = \frac{y^{2} - 5 x^{4} a}{x (x^{2} - 2 y)} - \frac{3 x y}{x^{2} - 2 y}$

Step 5.3.3.6

To write $- \frac{3 x y}{x^{2} - 2 y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$y' = \frac{y^{2} - 5 x^{4} a}{x (x^{2} - 2 y)} - \frac{3 x y}{x^{2} - 2 y} \cdot \frac{x}{x}$

Step 5.3.3.7

Write each expression with a common denominator of $x (x^{2} - 2 y)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.3.7.1

Multiply $\frac{3 x y}{x^{2} - 2 y}$ by $\frac{x}{x}$ .

$y' = \frac{y^{2} - 5 x^{4} a}{x (x^{2} - 2 y)} - \frac{3 x y x}{(x^{2} - 2 y) x}$

Step 5.3.3.7.2

Reorder the factors of $(x^{2} - 2 y) x$ .

$y' = \frac{y^{2} - 5 x^{4} a}{x (x^{2} - 2 y)} - \frac{3 x y x}{x (x^{2} - 2 y)}$

Step 5.3.3.8

Combine the numerators over the common denominator.

$y' = \frac{y^{2} - 5 x^{4} a - 3 x y x}{x (x^{2} - 2 y)}$

Step 5.3.3.9

Multiply $x$ by $x$ by adding the exponents.

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Step 5.3.3.9.1

Move $x$ .

$y' = \frac{y^{2} - 5 x^{4} a - 3 (x \cdot x) y}{x (x^{2} - 2 y)}$

Step 5.3.3.9.2

Multiply $x$ by $x$ .

$y' = \frac{y^{2} - 5 x^{4} a - 3 x^{2} y}{x (x^{2} - 2 y)}$

Step 6

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

$\frac{d y}{d x} = \frac{y^{2} - 5 x^{4} a - 3 x^{2} y}{x (x^{2} - 2 y)}$