tangent of f(x)=(x+2)/(x^2-9) at x=1

Check the domain before calculating the tangent:

x \in ℝ \ {-3, 3}

Find the tangent at

x = 1

⇥1) Let's differentiate:

\frac{d}{d x} (\frac{x + 2}{x^{2} - 9})

⇥We differentiate

p1

x + 2

and

q2

x^{2} - 9

using:

\frac{d}{d x} (\frac{p1}{q2}) = \frac{\frac{d}{d x} (p1) \cdot q2 - p1 \cdot \frac{d}{d x} (q2)}{{q2}^{2}}

⇥2) Let's perform the differentiation

p1

\frac{d}{d x} (x + 2)

⇥ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (x + 2)

1 + 0

⇥

1 + 0

⇥ⓘ Adding 0 changes nothing:

1 + 0

1

⇥

1

⇥3) Let's differentiate

q2

\frac{d}{d x} (x^{2} - 9)

⇥ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (x^{2} - 9)

\frac{d}{d x} (x^{2}) + 0

⇥

\frac{d}{d x} (x^{2}) + 0

⇥ⓘ Adding 0 changes nothing:

\frac{d}{d x} (x^{2}) + 0

\frac{d}{d x} (x^{2})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{2})

2 x

⇥

2 x

⇥4) We evaluate the resulting derivative (

\frac{p1}{q2}

)':

\frac{1 \cdot (x^{2} - 9) - (x + 2) \cdot 2 x}{{(x^{2} - 9)}^{2}}

⇥

\frac{x^{2} - 9 - (x + 2) \cdot 2 x}{{(x^{2} - 9)}^{2}}

⇥ⓘ Expand by multiplying each term inside the parentheses:

(x + 2) \cdot 2

x \cdot 2 + 4

⇥

\frac{x^{2} - 9 - (x \cdot 2 + 4) \cdot x}{{(x^{2} - 9)}^{2}}

⇥ⓘ Expand by multiplying each term inside the parentheses:

- (x \cdot 2 + 4) \cdot x

- (2 x^{2} + 4 x)

⇥

\frac{x^{2} - 9 - (2 x^{2} + 4 x)}{{(x^{2} - 9)}^{2}}

⇥ⓘ A minus sign before a parenthesis changes the sign of each of its terms:

- (2 x^{2} + 4 x)

- 2 x^{2} - 4 x

⇥

\frac{x^{2} - 9 - 2 x^{2} - 4 x}{{(x^{2} - 9)}^{2}}

⇥ⓘ Like terms are grouped by adding their coefficients:

x^{2} - 2 x^{2}

- x^{2}

⇥

\frac{- x^{2} - 9 - 4 x}{{(x^{2} - 9)}^{2}}

⇥ⓘ The minus sign is factored out, changing the sign of each term:

- x^{2} - 9 - 4 x

- (x^{2} + 9 + 4 x)

⇥

- \frac{x^{2} + 9 + 4 x}{{(x^{2} - 9)}^{2}}

⇥5) Then you can resume the differentiation:

⇥ⅰ

\frac{d}{d x} (\frac{x + 2}{x^{2} - 9})

- \frac{x^{2} + 9 + 4 x}{{(x^{2} - 9)}^{2}}

⇥

- \frac{x^{2} + 9 + 4 x}{{(x^{2} - 9)}^{2}}

⇥

- \frac{x^{2} + 4 x + 9}{{(x^{2} - 9)}^{2}}

Compute f(

1

)

⇥

\frac{1 + 2}{1^{2} - 9}

⇥ⓘ The numerator and the denominator are computed before the division:

1 + 2

3

1 - 9

-8

⇥

\frac{3}{-8}

⇥ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{3}{-8}

- \frac{3}{8}

⇥

- \frac{3}{8}

The point of tangency is (

1

- \frac{3}{8}

)

Compute f'(

1

)

⇥

- \frac{1^{2} + 4 \cdot 1 + 9}{{(1^{2} - 9)}^{2}}

⇥ⓘ Parentheses are computed first:

1 - 9

-8

⇥

- \frac{1 + 4 + 9}{{(-8)}^{2}}

⇥ⓘ The numerator and the denominator are computed before the division:

1 + 4

5

■Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-8)}^{2}

(-8) \cdot (-8)

64

⇥

- \frac{5 + 9}{64}

⇥ⓘ The numerator and the denominator are computed before the division:

5 + 9

14

⇥

- \frac{14}{64}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{14}{64}

- \frac{7 \cdot 2}{32 \cdot 2}

- \frac{7}{32}

⇥

- \frac{7}{32}

The slope is f'(

1

) =

- \frac{7}{32}

The tangent passes through this point with this slope:

y = - \frac{7}{32} \cdot (x - 1) - \frac{3}{8}

Compute the y-intercept:

b = - \frac{3}{8} + \frac{7}{32}

⇥

- \frac{3}{8} + \frac{7}{32}

⇥ⓘ To add fractions, put them over the same denominator, then add the numerators. We choose

32

, the least common multiple of

8

and

32

. Multiplying the numerator and the denominator by the same number does not change the fraction:

- \frac{3}{8} + \frac{7}{32}

\frac{-3 \cdot 4}{8 \cdot 4} + \frac{7}{32}

\frac{- 3 \cdot 4 + 7}{32}

⇥

\frac{- 3 \cdot 4 + 7}{32}

⇥ⓘ Multiplications are done before additions and subtractions:

3 \cdot 4

12

⇥

\frac{- 12 + 7}{32}

⇥ⓘ The numerator and the denominator are computed before the division:

- 12 + 7

-5

⇥

- \frac{5}{32}

⇥

y = - \frac{7}{32} \cdot x - \frac{5}{32}

⇥ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{7}{32} \cdot x

\frac{7 x}{32}

■To add fractions, put them over the same denominator, then add the numerators:

- \frac{7 x}{32} - \frac{5}{32}

\frac{- 7 x - 5}{32}

⇥

y = \frac{- 7 x - 5}{32}

ⓘ The minus sign is factored out, changing the sign of each term:

- 7 x - 5

- (7 x + 5)

▶Tangent line:

y = - \frac{7 x + 5}{32}