tangent of f(x)=1/x at x=-2 — step-by-step solution

Check the domain before calculating the tangent:

x \neq 0

Find the tangent at

x = -2

To ease differentiation, the expression

\frac{1}{x}

is turned into

x^{-1}

Let's perform the differentiation:

\frac{d}{d x} (x^{-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

ⓘ The derivative of x^n is n*x^(n-1):

\frac{d}{d x} (x^{-1})

-1 \cdot x^{-2}

-1 \cdot x^{-2}

ⓘ Multiplying by -1 gives the opposite:

-1 \cdot x^{-2}

- x^{-2}

- x^{-2}

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

- x^{-2}

- \frac{1}{x^{2}}

- \frac{1}{x^{2}}

Compute f(

-2

)

\frac{1}{-2}

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

\frac{1}{-2}

- \frac{1}{2}

- \frac{1}{2}

The point of tangency is (

-2

- \frac{1}{2}

)

Compute f'(

-2

)

- \frac{1}{{(-2)}^{2}}

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

{(-2)}^{2}

(-2) \cdot (-2)

4

- \frac{1}{4}

The slope is f'(

-2

) =

- \frac{1}{4}

The tangent passes through this point with this slope:

y = - \frac{1}{4} \cdot (x + 2) - \frac{1}{2}

Compute the y-intercept:

b = - \frac{1}{2} - (- \frac{1}{4} \cdot (-2))

- \frac{1}{2} - (- \frac{1}{4} \cdot (-2))

- \frac{1}{2} - \frac{1}{4} \cdot 2

ⓘ Multiplying a fraction by a number multiplies its numerator:

\frac{1}{4} \cdot 2

\frac{1 \cdot 2}{4}

\frac{2}{4}

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

4

, the least common multiple of

2

and

4

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

- \frac{1}{2} - \frac{2}{4}

\frac{-1 \cdot 2}{2 \cdot 2} - \frac{2}{4}

\frac{- 1 \cdot 2 - 2}{4}

\frac{- 1 \cdot 2 - 2}{4}

ⓘ Multiplications are done before additions and subtractions:

1 \cdot 2

2

\frac{- 2 - 2}{4}

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

- 2 - 2

-4

- \frac{4}{4}

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

4

, their greatest common divisor:

- \frac{4}{4}

-1

-1

y = - \frac{1}{4} \cdot x - 1

ⓘ Multiplying a fraction by a term multiplies its numerator:

\frac{1}{4} \cdot x

\frac{1 \cdot x}{4}

\frac{x}{4}

▶Tangent line:

y = - \frac{x}{4} - 1