tangent of (64)/(x^2+16) at x=3 — step-by-step solution

Check the domain before calculating the tangent:

x \in ℝ

Find the tangent at

x = 3

To ease differentiation, the expression

\frac{64}{x^{2} + 16}

is turned into

64 \cdot {(x^{2} + 16)}^{-1}

Let's differentiate:

\frac{d}{d x} (64 \cdot {(x^{2} + 16)}^{-1})

Apply the differentiation rule:

\frac{d}{d x} (f^{n}) = n \cdot \frac{d}{d x} (f) \cdot f^{n - 1}

with f =

x^{2} + 16

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

\frac{d}{d x} (64 \cdot {(x^{2} + 16)}^{-1})

64 \cdot \frac{d}{d x} ({(x^{2} + 16)}^{-1})

■The derivative of x^n is n*x^(n-1):

\frac{d}{d x} ({(x^{2} + 16)}^{-1})

-1 \cdot \frac{d}{d x} (x^{2} + 16) \cdot {(x^{2} + 16)}^{-2}

64 \cdot (-1 \cdot \frac{d}{d x} (x^{2} + 16) \cdot {(x^{2} + 16)}^{-2})

ⓘ A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

64 \cdot (-1)

-64

■The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (x^{2} + 16)

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

-64 \cdot (\frac{d}{d x} (x^{2}) + 0) \cdot {(x^{2} + 16)}^{-2}

ⓘ Adding 0 changes nothing:

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

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

■The factor is moved to the numerator or the denominator by changing the sign of its exponent, for a nonzero base:

- 64 \cdot \frac{d}{d x} (x^{2}) \cdot {(x^{2} + 16)}^{-2}

- \frac{64 \cdot \frac{d}{d x} (x^{2})}{{(x^{2} + 16)}^{2}}

- \frac{64 \cdot \frac{d}{d x} (x^{2})}{{(x^{2} + 16)}^{2}}

ⓘ The derivative of x^n is n*x^(n-1):

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

2 x

- \frac{64 \cdot 2 x}{{(x^{2} + 16)}^{2}}

ⅰ

64 \cdot 2

(60 + 4) \cdot 2

60 \cdot 2 + 4 \cdot 2

120 + 8

128

- \frac{128 x}{{(x^{2} + 16)}^{2}}

Compute f(

3

)

\frac{64}{3^{2} + 16}

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

3^{2}

3 \cdot 3

9

\frac{64}{9 + 16}

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

9 + 16

25

\frac{64}{25}

The point of tangency is (

3

\frac{64}{25}

)

Compute f'(

3

)

- \frac{128 \cdot 3}{{(3^{2} + 16)}^{2}}

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

3^{2}

3 \cdot 3

9

ⅰ

128 \cdot 3

(100 + 20 + 8) \cdot 3

100 \cdot 3 + 20 \cdot 3 + 8 \cdot 3

300 + 60 + 24

384

- \frac{384}{{(9 + 16)}^{2}}

ⓘ Parentheses are computed first:

9 + 16

25

- \frac{384}{25^{2}}

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

25^{2}

25 \cdot 25

625

- \frac{384}{625}

The slope is f'(

3

) =

- \frac{384}{625}

The tangent passes through this point with this slope:

y = - \frac{384}{625} \cdot (x - 3) + \frac{64}{25}

Compute the y-intercept:

b = \frac{64}{25} - (- \frac{384}{625} \cdot 3)

\frac{64}{25} - (- \frac{384}{625} \cdot 3)

ⓘ Two minus signs cancel out, the product of two negative numbers is positive:

- (- \frac{384}{625} \cdot 3)

\frac{384}{625} \cdot 3

\frac{64}{25} + \frac{384}{625} \cdot 3

ⓘ Multiplying a fraction by a number multiplies its numerator:

\frac{384}{625} \cdot 3

\frac{384 \cdot 3}{625}

\frac{1152}{625}

\frac{64}{25} + \frac{1152}{625}

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

625

, the least common multiple of

25

and

625

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

\frac{64}{25} + \frac{1152}{625}

\frac{64 \cdot 25}{25 \cdot 25} + \frac{1152}{625}

\frac{64 \cdot 25 + 1152}{625}

\frac{64 \cdot 25 + 1152}{625}

ⓘ Multiplications are done before additions and subtractions:

64 \cdot 25

64 \cdot (20 + 5)

64 \cdot 20 + 64 \cdot 5

1280 + 320

1600

\frac{1600 + 1152}{625}

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

1600 + 1152

2752

\frac{2752}{625}

y = - \frac{384}{625} \cdot x + \frac{2752}{625}

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{384}{625} \cdot x

\frac{384 x}{625}

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

- \frac{384 x}{625} + \frac{2752}{625}

\frac{- 384 x + 2752}{625}

▶Tangent line:

y = \frac{- 384 x + 2752}{625}