tangent of y=(5x)/(x+2),(3,3) — step-by-step solution

Check the domain before calculating the tangent:

x \neq -2

Find the tangent at

x = 3

⇥1) Let's differentiate:

\frac{d}{d x} (\frac{5 x}{x + 2})

⇥For

p1

5 x

and

q2

x + 2

, use the differentiation rule:

\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 differentiate

p1

\frac{d}{d x} (5 x)

⇥ⓘ A constant factor stays in front of the derivative, (k*u)' = k*u':

\frac{d}{d x} (5 x)

5

⇥

5

⇥3) Let's perform the differentiation

q2

\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

⇥4) Let's evaluate the resulting derivative (

\frac{p1}{q2}

)':

\frac{5 \cdot (x + 2) - 5 x \cdot 1}{{(x + 2)}^{2}}

⇥

\frac{5 \cdot (x + 2) - 5 x}{{(x + 2)}^{2}}

⇥ⓘ Expand by multiplying each term inside the parentheses:

5 \cdot (x + 2)

5 x + 10

⇥

\frac{5 x + 10 - 5 x}{{(x + 2)}^{2}}

⇥ⓘ Like terms are grouped by adding their coefficients:

5 x - 5 x

0

⇥

\frac{0 + 10}{{(x + 2)}^{2}}

⇥ⓘ Adding 0 changes nothing:

0 + 10

10

⇥

\frac{10}{{(x + 2)}^{2}}

⇥5) You can then resume the differentiation:

⇥ⅰ

\frac{d}{d x} (\frac{5 x}{x + 2})

\frac{10}{{(x + 2)}^{2}}

⇥

\frac{10}{{(x + 2)}^{2}}

Compute f(

3

)

⇥

\frac{5 \cdot 3}{3 + 2}

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

3 + 2

5

⇥ⅰ

5 \cdot 3

15

⇥

\frac{15}{5}

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

5

, their greatest common divisor:

\frac{15}{5}

3

⇥

3

The point of tangency is (

3

3

)

Compute f'(

3

)

⇥

\frac{10}{{(3 + 2)}^{2}}

⇥ⓘ Parentheses are computed first:

3 + 2

5

⇥

\frac{10}{5^{2}}

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

5^{2}

5 \cdot 5

25

⇥

\frac{10}{25}

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

5

, their greatest common divisor:

\frac{10}{25}

\frac{2 \cdot 5}{5 \cdot 5}

\frac{2}{5}

⇥

\frac{2}{5}

The slope is f'(

3

) =

\frac{2}{5}

The tangent passes through this point with this slope:

y = \frac{2}{5} \cdot (x - 3) + 3

Compute the y-intercept:

b = 3 - \frac{2}{5} \cdot 3

⇥

3 - \frac{2}{5} \cdot 3

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

\frac{2}{5} \cdot 3

\frac{2 \cdot 3}{5}

\frac{6}{5}

⇥

3 - \frac{6}{5}

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

3 - \frac{6}{5}

\frac{5 \cdot 3 - 6}{5}

⇥

\frac{5 \cdot 3 - 6}{5}

⇥ⓘ Multiplications are done before additions and subtractions:

5 \cdot 3

15

⇥

\frac{15 - 6}{5}

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

15 - 6

9

⇥

\frac{9}{5}

⇥

y = \frac{2}{5} \cdot x + \frac{9}{5}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{2}{5} \cdot x

\frac{2 x}{5}

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

\frac{2 x}{5} + \frac{9}{5}

\frac{2 x + 9}{5}

▶Tangent line:

y = \frac{2 x + 9}{5}