Calculus Examples

$f (x) = \frac{y - 3}{y^{2} - 3 y + 9}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

Find the first derivative.

Tap for more steps...

Step 1.1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = y - 3$ and $g (y) = y^{2} - 3 y + 9$ .

$f' (y) = \frac{(y^{2} - 3 y + 9) \frac{d}{d y} (y - 3) - (y - 3) \frac{d}{d y} (y^{2} - 3 y + 9)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

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

$f' (y) = \frac{(y^{2} - 3 y + 9) (\frac{d}{d y} (y) + \frac{d}{d y} (- 3)) - (y - 3) \frac{d}{d y} (y^{2} - 3 y + 9)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2.2

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

$f' (y) = \frac{(y^{2} - 3 y + 9) (1 + \frac{d}{d y} (- 3)) - (y - 3) \frac{d}{d y} (y^{2} - 3 y + 9)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2.3

Since $- 3$ is constant with respect to $y$ , the derivative of $- 3$ with respect to $y$ is $0$ .

$f' (y) = \frac{(y^{2} - 3 y + 9) (1 + 0) - (y - 3) \frac{d}{d y} (y^{2} - 3 y + 9)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2.4

Simplify the expression.

Tap for more steps...

Step 1.1.2.4.1

Add $1$ and $0$ .

$f' (y) = \frac{(y^{2} - 3 y + 9) \cdot 1 - (y - 3) \frac{d}{d y} (y^{2} - 3 y + 9)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2.4.2

Multiply $y^{2} - 3 y + 9$ by $1$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - (y - 3) \frac{d}{d y} (y^{2} - 3 y + 9)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2.5

By the Sum Rule, the derivative of $y^{2} - 3 y + 9$ with respect to $y$ is $\frac{d}{d y} [y^{2}] + \frac{d}{d y} [- 3 y] + \frac{d}{d y} [9]$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - (y - 3) (\frac{d}{d y} (y^{2}) + \frac{d}{d y} (- 3 y) + \frac{d}{d y} (9))}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2.6

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

$f' (y) = \frac{y^{2} - 3 y + 9 - (y - 3) (2 y + \frac{d}{d y} (- 3 y) + \frac{d}{d y} (9))}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2.7

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

$f' (y) = \frac{y^{2} - 3 y + 9 - (y - 3) (2 y - 3 \frac{d}{d y} y + \frac{d}{d y} (9))}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2.8

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

$f' (y) = \frac{y^{2} - 3 y + 9 - (y - 3) (2 y - 3 \cdot 1 + \frac{d}{d y} (9))}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2.9

Multiply $- 3$ by $1$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - (y - 3) (2 y - 3 + \frac{d}{d y} (9))}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2.10

Since $9$ is constant with respect to $y$ , the derivative of $9$ with respect to $y$ is $0$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - (y - 3) (2 y - 3 + 0)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.2.11

Add $2 y - 3$ and $0$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - (y - 3) (2 y - 3)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3

Simplify.

Tap for more steps...

Step 1.1.3.1

Apply the distributive property.

$f' (y) = \frac{y^{2} - 3 y + 9 + (- y + 3) (2 y - 3)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2

Simplify the numerator.

Tap for more steps...

Step 1.1.3.2.1

Simplify each term.

Tap for more steps...

Step 1.1.3.2.1.1

Multiply $- 1$ by $- 3$ .

$f' (y) = \frac{y^{2} - 3 y + 9 + (- y + 3) (2 y - 3)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.1.2

Expand $(- y + 3) (2 y - 3)$ using the FOIL Method.

Tap for more steps...

Step 1.1.3.2.1.2.1

Apply the distributive property.

$f' (y) = \frac{y^{2} - 3 y + 9 - y (2 y - 3) + 3 (2 y - 3)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.1.2.2

Apply the distributive property.

$f' (y) = \frac{y^{2} - 3 y + 9 - y (2 y) - y \cdot - 3 + 3 (2 y - 3)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.1.2.3

Apply the distributive property.

$f' (y) = \frac{y^{2} - 3 y + 9 - y (2 y) - y \cdot - 3 + 3 (2 y) + 3 \cdot - 3}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.1.3.2.1.3.1

Simplify each term.

Tap for more steps...

Step 1.1.3.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$f' (y) = \frac{y^{2} - 3 y + 9 - 1 \cdot (2 y \cdot y) - y \cdot - 3 + 3 (2 y) + 3 \cdot - 3}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.1.3.1.2

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 1.1.3.2.1.3.1.2.1

Move $y$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - 1 \cdot (2 (y \cdot y)) - y \cdot - 3 + 3 (2 y) + 3 \cdot - 3}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.1.3.1.2.2

Multiply $y$ by $y$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - 1 \cdot (2 y^{2}) - y \cdot - 3 + 3 (2 y) + 3 \cdot - 3}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.1.3.1.3

Multiply $- 1$ by $2$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - 2 y^{2} - y \cdot - 3 + 3 (2 y) + 3 \cdot - 3}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.1.3.1.4

Multiply $- 3$ by $- 1$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - 2 y^{2} + 3 y + 3 (2 y) + 3 \cdot - 3}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.1.3.1.5

Multiply $2$ by $3$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - 2 y^{2} + 3 y + 6 y + 3 \cdot - 3}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.1.3.1.6

Multiply $3$ by $- 3$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - 2 y^{2} + 3 y + 6 y - 9}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.1.3.2

Add $3 y$ and $6 y$ .

$f' (y) = \frac{y^{2} - 3 y + 9 - 2 y^{2} + 9 y - 9}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.2

Combine the opposite terms in $y^{2} - 3 y + 9 - 2 y^{2} + 9 y - 9$ .

Tap for more steps...

Step 1.1.3.2.2.1

Subtract $9$ from $9$ .

$f' (y) = \frac{y^{2} - 3 y - 2 y^{2} + 9 y + 0}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.2.2

Add $y^{2} - 3 y - 2 y^{2} + 9 y$ and $0$ .

$f' (y) = \frac{y^{2} - 3 y - 2 y^{2} + 9 y}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.3

Subtract $2 y^{2}$ from $y^{2}$ .

$f' (y) = \frac{- y^{2} - 3 y + 9 y}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.2.4

Add $- 3 y$ and $9 y$ .

$f' (y) = \frac{- y^{2} + 6 y}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.3

Factor $y$ out of $- y^{2} + 6 y$ .

Tap for more steps...

Step 1.1.3.3.1

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

$f' (y) = \frac{y (- y) + 6 y}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.3.2

Factor $y$ out of $6 y$ .

$f' (y) = \frac{y (- y) + y \cdot 6}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.3.3

Factor $y$ out of $y (- y) + y \cdot 6$ .

$f' (y) = \frac{y (- y + 6)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.4

Factor $- 1$ out of $- y$ .

$f' (y) = \frac{y (- (y) + 6)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.5

Rewrite $6$ as $- 1 (- 6)$ .

$f' (y) = \frac{y (- (y) - 1 \cdot - 6)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.6

Factor $- 1$ out of $- (y) - 1 (- 6)$ .

$f' (y) = \frac{y (- (y - 6))}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.7

Rewrite $- (y - 6)$ as $- 1 (y - 6)$ .

$f' (y) = \frac{y (- 1 (y - 6))}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.1.3.8

Move the negative in front of the fraction.

$f' (y) = - \frac{y (y - 6)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{y (y - 6)}{{(y^{2} - 3 y + 9)}^{2}}$ .

$- \frac{y (y - 6)}{{(y^{2} - 3 y + 9)}^{2}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{y (y - 6)}{{(y^{2} - 3 y + 9)}^{2}} = 0$ .

Tap for more steps...

Step 2.1

Set the first derivative equal to $0$ .

$- \frac{y (y - 6)}{{(y^{2} - 3 y + 9)}^{2}} = 0$

Step 2.2

Set the numerator equal to zero.

$y (y - 6) = 0$

Step 2.3

Solve the equation for $x$ .

Tap for more steps...

Step 2.3.1

Simplify $y (y - 6)$ .

Tap for more steps...

Step 2.3.1.1

Apply the distributive property.

$y \cdot y + y \cdot - 6 = 0$

Step 2.3.1.2

Simplify the expression.

Tap for more steps...

Step 2.3.1.2.1

Multiply $y$ by $y$ .

$y^{2} + y \cdot - 6 = 0$

Step 2.3.1.2.2

Move $- 6$ to the left of $y$ .

$y^{2} - 6 y = 0$

Step 2.3.2

Factor $y$ out of $y^{2} - 6 y$ .

Tap for more steps...

Step 2.3.2.1

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

$y \cdot y - 6 y = 0$

Step 2.3.2.2

Factor $y$ out of $- 6 y$ .

$y \cdot y + y \cdot - 6 = 0$

Step 2.3.2.3

Factor $y$ out of $y \cdot y + y \cdot - 6$ .

$y (y - 6) = 0$

Step 2.3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y = 0$

$y - 6 = 0$

Step 2.3.4

Set $y$ equal to $0$ .

$y = 0$

Step 2.3.5

Set $y - 6$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 2.3.5.1

Set $y - 6$ equal to $0$ .

$y - 6 = 0$

Step 2.3.5.2

Add $6$ to both sides of the equation.

$y = 6$

Step 2.3.6

The final solution is all the values that make $y (y - 6) = 0$ true.

$y = 0, 6$

Step 3

Evaluate $\frac{y - 3}{y^{2} - 3 y + 9}$ at each $x$ value where the derivative is $0$ or undefined.

Tap for more steps...

Step 3.1

$\frac{y - 3}{y^{2} - 3 y + 9}$ is constant with respect to $x$ .

$\frac{y - 3}{y^{2} - 3 y + 9}$

Step 3.2

List all of the points.

$(0, \frac{y - 3}{y^{2} - 3 y + 9}), (6, \frac{y - 3}{y^{2} - 3 y + 9})$

Step 4