Calculus Examples

$x y = 1$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (x y) = \frac{d}{d x} (1)$

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 $f (x) = x$ and $g (x) = y$ .

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

Step 2.2

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

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

Step 2.3

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

$x y' + y \cdot 1$

Step 2.4

Multiply $y$ by $1$ .

$x y' + y$

Step 3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0$

Step 4

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

$x y' + y = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $y$ from both sides of the equation.

$x y' = - y$

Step 5.2

Divide each term in $x y' = - y$ by $x$ and simplify.

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Step 5.2.1

Divide each term in $x y' = - y$ by $x$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $x$ .

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Step 5.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.1.2

Divide $y'$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 7.1

Find the LCD of the terms in the equation.

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Step 7.1.1

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

$x, 1$

Step 7.1.2

The LCM of one and any expression is the expression.

$x$

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $x$ .

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Step 7.2.2.1.1

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

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

Step 7.2.2.1.2

Cancel the common factor.

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

Step 7.2.2.1.3

Rewrite the expression.

$- y = 0 x$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Multiply $0$ by $x$ .

$- y = 0$

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

Simplify the left side.

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Step 7.3.2.1

Dividing two negative values results in a positive value.

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

Step 7.3.2.2

Divide $y$ by $1$ .

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

Step 7.3.3

Simplify the right side.

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Step 7.3.3.1

Divide $0$ by $- 1$ .

$y = 0$

Step 7.4

The variable $x$ got canceled.

All real numbers

Step 8

Solve for $y$ .

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Step 8.1

Simplify.

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Step 8.1.1

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

$A (l^{1} l) (r e a l) (n u m b e r s) y = 1$

Step 8.1.2

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

$A (l^{1} l^{1}) (r e a l) (n u m b e r s) y = 1$

Step 8.1.3

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

$A l^{1 + 1} (r e a l) (n u m b e r s) y = 1$

Step 8.1.4

Add $1$ and $1$ .

$A l^{2} (r e a l) (n u m b e r s) y = 1$

Step 8.1.5

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

$A (l^{1} l^{2}) (r e a) (n u m b e r s) y = 1$

Step 8.1.6

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

$A l^{1 + 2} (r e a) (n u m b e r s) y = 1$

Step 8.1.7

Add $1$ and $2$ .

$A l^{3} (r e a) (n u m b e r s) y = 1$

Step 8.1.8

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

$A l^{3} (e a) (n u m b e (r^{1} r) s) y = 1$

Step 8.1.9

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

$A l^{3} (e a) (n u m b e (r^{1} r^{1}) s) y = 1$

Step 8.1.10

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

$A l^{3} (e a) (n u m b e r^{1 + 1} s) y = 1$

Step 8.1.11

Add $1$ and $1$ .

$A l^{3} (e a) (n u m b e r^{2} s) y = 1$

Step 8.1.12

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

$A l^{3} (a) (n u m b (e^{1} e) r^{2} s) y = 1$

Step 8.1.13

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

$A l^{3} (a) (n u m b (e^{1} e^{1}) r^{2} s) y = 1$

Step 8.1.14

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

$A l^{3} (a) (n u m b e^{1 + 1} r^{2} s) y = 1$

Step 8.1.15

Add $1$ and $1$ .

$A l^{3} (a) (n u m b e^{2} r^{2} s) y = 1$

Step 8.1.16

Multiply $A l^{3}$ by $a$ .

$A l^{3} a (n u m b e^{2} r^{2} s) y = 1$

Step 8.1.17

Remove parentheses.

$A l^{3} a (n u m b e^{2} r^{2}) s y = 1$

Step 8.1.18

Remove parentheses.

$A l^{3} a (n u m b e^{2}) r^{2} s y = 1$

Step 8.1.19

Remove parentheses.

$A l^{3} a (n u m b) e^{2} r^{2} s y = 1$

Step 8.1.20

Remove parentheses.

$A l^{3} a (n u m) b e^{2} r^{2} s y = 1$

Step 8.1.21

Remove parentheses.

$A l^{3} a (n u) m b e^{2} r^{2} s y = 1$

Step 8.1.22

Remove parentheses.

$A l^{3} a n u m b e^{2} r^{2} s y = 1$

Step 8.2

Divide each term in $A l^{3} a n u m b e^{2} r^{2} s y = 1$ by $A l^{3} a n u m b e^{2} r^{2} s$ and simplify.

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Step 8.2.1

Divide each term in $A l^{3} a n u m b e^{2} r^{2} s y = 1$ by $A l^{3} a n u m b e^{2} r^{2} s$ .

$\frac{A l^{3} a n u m b e^{2} r^{2} s y}{A l^{3} a n u m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $A$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{A l^{3} a n u m b e^{2} r^{2} s y}{A l^{3} a n u m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.1.2

Rewrite the expression.

$\frac{l^{3} a n u m b e^{2} r^{2} s y}{l^{3} a n u m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.2

Cancel the common factor of $l^{3}$ .

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Step 8.2.2.2.1

Cancel the common factor.

$\frac{l^{3} a n u m b e^{2} r^{2} s y}{l^{3} a n u m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.2.2

Rewrite the expression.

$\frac{a n u m b e^{2} r^{2} s y}{a n u m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.3

Cancel the common factor of $a$ .

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Step 8.2.2.3.1

Cancel the common factor.

$\frac{a n u m b e^{2} r^{2} s y}{a n u m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.3.2

Rewrite the expression.

$\frac{n u m b e^{2} r^{2} s y}{n u m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.4

Cancel the common factor of $n$ .

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Step 8.2.2.4.1

Cancel the common factor.

$\frac{n u m b e^{2} r^{2} s y}{n u m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.4.2

Rewrite the expression.

$\frac{u m b e^{2} r^{2} s y}{u m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.5

Cancel the common factor of $u$ .

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Step 8.2.2.5.1

Cancel the common factor.

$\frac{u m b e^{2} r^{2} s y}{u m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.5.2

Rewrite the expression.

$\frac{m b e^{2} r^{2} s y}{m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.6

Cancel the common factor of $m$ .

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Step 8.2.2.6.1

Cancel the common factor.

$\frac{m b e^{2} r^{2} s y}{m b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.6.2

Rewrite the expression.

$\frac{b e^{2} r^{2} s y}{b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.7

Cancel the common factor of $b$ .

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Step 8.2.2.7.1

Cancel the common factor.

$\frac{b e^{2} r^{2} s y}{b e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.7.2

Rewrite the expression.

$\frac{e^{2} r^{2} s y}{e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.8

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

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Step 8.2.2.8.1

Cancel the common factor.

$\frac{e^{2} r^{2} s y}{e^{2} r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.8.2

Rewrite the expression.

$\frac{r^{2} s y}{r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.9

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

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Step 8.2.2.9.1

Cancel the common factor.

$\frac{r^{2} s y}{r^{2} s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.9.2

Rewrite the expression.

$\frac{s y}{s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.10

Cancel the common factor of $s$ .

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Step 8.2.2.10.1

Cancel the common factor.

$\frac{s y}{s} = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 8.2.2.10.2

Divide $y$ by $1$ .

$y = \frac{1}{A l^{3} a n u m b e^{2} r^{2} s}$

Step 9

Find the points where $\frac{d y}{d x} = 0$ .

$(A l \cdot l (r e a l) (n u m b e r s), \frac{1}{A l^{3} a n u m b e^{2} r^{2} s})$

Step 10