Calculus Examples

$x^{2} y^{2} = 36$

Step 1

Solve the equation as $y$ in terms of $x$ .

Tap for more steps...

Step 1.1

Divide each term in $x^{2} y^{2} = 36$ by $x^{2}$ and simplify.

Tap for more steps...

Step 1.1.1

Divide each term in $x^{2} y^{2} = 36$ by $x^{2}$ .

$\frac{x^{2} y^{2}}{x^{2}} = \frac{36}{x^{2}}$

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

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

Tap for more steps...

Step 1.1.2.1.1

Cancel the common factor.

$\frac{x^{2} y^{2}}{x^{2}} = \frac{36}{x^{2}}$

Step 1.1.2.1.2

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

$y^{2} = \frac{36}{x^{2}}$

Step 1.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{\frac{36}{x^{2}}}$

Step 1.3

Simplify $\pm \sqrt{\frac{36}{x^{2}}}$ .

Tap for more steps...

Step 1.3.1

Rewrite $36$ as $6^{2}$ .

$y = \pm \sqrt{\frac{6^{2}}{x^{2}}}$

Step 1.3.2

Rewrite $\frac{6^{2}}{x^{2}}$ as ${(\frac{6}{x})}^{2}$ .

$y = \pm \sqrt{{(\frac{6}{x})}^{2}}$

Step 1.3.3

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm \frac{6}{x}$

Step 1.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 1.4.1

First, use the positive value of the $\pm$ to find the first solution.

$y = \frac{6}{x}$

Step 1.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$y = - \frac{6}{x}$

Step 1.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$y = \frac{6}{x}$

$y = - \frac{6}{x}$

$y = \frac{6}{x}$

$y = - \frac{6}{x}$

$y = \frac{6}{x}$

$y = - \frac{6}{x}$

Step 2

Set each solution of $y$ as a function of $x$ .

$y = \frac{6}{x} \to f (x) = \frac{6}{x}$

$y = - \frac{6}{x} \to f (x) = - \frac{6}{x}$

Step 3

Because the $y$ variable in the equation $x^{2} y^{2} = 36$ has a degree greater than $1$ , use implicit differentiation to solve for the derivative $\frac{d y}{d x}$ .

Tap for more steps...

Step 3.1

Differentiate both sides of the equation.

$\frac{d}{d x} (x^{2} y^{2}) = \frac{d}{d x} (36)$

Step 3.2

Differentiate the left side of the equation.

Tap for more steps...

Step 3.2.1

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

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

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 $g (x) = y$ .

Tap for more steps...

Step 3.2.2.1

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

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

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 $n = 2$ .

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

Step 3.2.2.3

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

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

Step 3.2.3

Move $2$ to the left of $x^{2}$ .

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

Step 3.2.4

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

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

Step 3.2.5

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

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

Step 3.2.6

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

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

Step 3.3

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$0$

Step 3.4

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

$2 x^{2} y y' + 2 y^{2} x = 0$

Step 3.5

Solve for $y'$ .

Tap for more steps...

Step 3.5.1

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

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

Step 3.5.2

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

Tap for more steps...

Step 3.5.2.1

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

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

Step 3.5.2.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.5.2.2.1.1

Cancel the common factor.

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

Step 3.5.2.2.1.2

Rewrite the expression.

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

Step 3.5.2.2.2

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

Tap for more steps...

Step 3.5.2.2.2.1

Cancel the common factor.

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

Step 3.5.2.2.2.2

Rewrite the expression.

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

Step 3.5.2.2.3

Cancel the common factor of $y$ .

Tap for more steps...

Step 3.5.2.2.3.1

Cancel the common factor.

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

Step 3.5.2.2.3.2

Divide $y'$ by $1$ .

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

Step 3.5.2.3

Simplify the right side.

Tap for more steps...

Step 3.5.2.3.1

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

Tap for more steps...

Step 3.5.2.3.1.1

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

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

Step 3.5.2.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.5.2.3.1.2.1

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

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

Step 3.5.2.3.1.2.2

Cancel the common factor.

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

Step 3.5.2.3.1.2.3

Rewrite the expression.

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

Step 3.5.2.3.2

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

Tap for more steps...

Step 3.5.2.3.2.1

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

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

Step 3.5.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 3.5.2.3.2.2.1

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

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

Step 3.5.2.3.2.2.2

Cancel the common factor.

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

Step 3.5.2.3.2.2.3

Rewrite the expression.

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

Step 3.5.2.3.3

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

Tap for more steps...

Step 3.5.2.3.3.1

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

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

Step 3.5.2.3.3.2

Cancel the common factors.

Tap for more steps...

Step 3.5.2.3.3.2.1

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

$y' = \frac{x (- y)}{x \cdot x}$

Step 3.5.2.3.3.2.2

Cancel the common factor.

$y' = \frac{x (- y)}{x \cdot x}$

Step 3.5.2.3.3.2.3

Rewrite the expression.

$y' = \frac{- y}{x}$

Step 3.5.2.3.4

Move the negative in front of the fraction.

$y' = - \frac{y}{x}$

Step 3.6

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

$\frac{d y}{d x} = - \frac{y}{x}$

Step 4

Set the derivative equal to $0$ then solve the equation $- \frac{y}{x} = 0$ .

Tap for more steps...

Step 4.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 4.1.2

The LCM of one and any expression is the expression.

$x$

Step 4.2

Multiply each term in $- \frac{y}{x} = 0$ by $x$ to eliminate the fractions.

Tap for more steps...

Step 4.2.1

Multiply each term in $- \frac{y}{x} = 0$ by $x$ .

$- \frac{y}{x} x = 0 x$

Step 4.2.2

Simplify the left side.

Tap for more steps...

Step 4.2.2.1

Cancel the common factor of $x$ .

Tap for more steps...

Step 4.2.2.1.1

Move the leading negative in $- \frac{y}{x}$ into the numerator.

$\frac{- y}{x} x = 0 x$

Step 4.2.2.1.2

Cancel the common factor.

$\frac{- y}{x} x = 0 x$

Step 4.2.2.1.3

Rewrite the expression.

$- y = 0 x$

Step 4.2.3

Simplify the right side.

Tap for more steps...

Step 4.2.3.1

Multiply $0$ by $x$ .

$- y = 0$

Step 4.3

Divide each term in $- y = 0$ by $- 1$ and simplify.

Tap for more steps...

Step 4.3.1

Divide each term in $- y = 0$ by $- 1$ .

$\frac{- y}{- 1} = \frac{0}{- 1}$

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{0}{- 1}$

Step 4.3.2.2

Divide $y$ by $1$ .

$y = \frac{0}{- 1}$

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Divide $0$ by $- 1$ .

$y = 0$

Step 4.4

The variable $x$ got canceled.

All real numbers

Step 5

Solve the function $f (x) = \frac{6}{x}$ at $x = A l \cdot l (r e a l) (n u m b e r s)$ .

Tap for more steps...

Step 5.1

Replace the variable $x$ with All real numbers in the expression.

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Multiply $l$ by $l$ by adding the exponents.

Tap for more steps...

Step 5.2.1.1

Move $l$ .

Step 5.2.1.2

Multiply $l$ by $l$ .

Step 5.2.2

Multiply $l^{2}$ by $l$ by adding the exponents.

Tap for more steps...

Step 5.2.2.1

Move $l$ .

Step 5.2.2.2

Multiply $l$ by $l^{2}$ .

Tap for more steps...

Step 5.2.2.2.1

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

Step 5.2.2.2.2

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

Step 5.2.2.3

Add $1$ and $2$ .

Step 5.2.3

Multiply $r$ by $r$ by adding the exponents.

Tap for more steps...

Step 5.2.3.1

Move $r$ .

Step 5.2.3.2

Multiply $r$ by $r$ .

Step 5.2.4

Simplify the denominator.

Tap for more steps...

Step 5.2.4.1

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

Step 5.2.4.2

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

Step 5.2.4.3

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

Step 5.2.4.4

Add $1$ and $1$ .

Step 5.2.5

The final answer is $\frac{6}{A l^{3} r^{2} a n u m b e^{2} s}$ .

$\frac{6}{A l^{3} r^{2} a n u m b e^{2} s}$

Step 6

Solve the function $f (x) = - \frac{6}{x}$ at $x = A l \cdot l (r e a l) (n u m b e r s)$ .

Tap for more steps...

Step 6.1

Replace the variable $x$ with All real numbers in the expression.

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Multiply $l$ by $l$ by adding the exponents.

Tap for more steps...

Step 6.2.1.1

Move $l$ .

Step 6.2.1.2

Multiply $l$ by $l$ .

Step 6.2.2

Multiply $l^{2}$ by $l$ by adding the exponents.

Tap for more steps...

Step 6.2.2.1

Move $l$ .

Step 6.2.2.2

Multiply $l$ by $l^{2}$ .

Tap for more steps...

Step 6.2.2.2.1

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

Step 6.2.2.2.2

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

Step 6.2.2.3

Add $1$ and $2$ .

Step 6.2.3

Multiply $r$ by $r$ by adding the exponents.

Tap for more steps...

Step 6.2.3.1

Move $r$ .

Step 6.2.3.2

Multiply $r$ by $r$ .

Step 6.2.4

Simplify the denominator.

Tap for more steps...

Step 6.2.4.1

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

Step 6.2.4.2

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

Step 6.2.4.3

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

Step 6.2.4.4

Add $1$ and $1$ .

Step 6.2.5

The final answer is $- \frac{6}{A l^{3} r^{2} a n u m b e^{2} s}$ .

$- \frac{6}{A l^{3} r^{2} a n u m b e^{2} s}$

Step 7

The horizontal tangent lines are $y = \frac{6}{A l^{3} r^{2} a n u m b e^{2} s}, y = - \frac{6}{A l^{3} r^{2} a n u m b e^{2} s}$

$y = \frac{6}{A l^{3} r^{2} a n u m b e^{2} s}, y = - \frac{6}{A l^{3} r^{2} a n u m b e^{2} s}$

Step 8