tangent of f(x)=3/(x^2) at x=1 — step-by-step solution

Check the domain before calculating the tangent:

x \neq 0

Find the tangent at

x = 1

To simplify differentiation, we rewrite

\frac{3}{x^{2}}

3 x^{-2}

Let's perform the differentiation:

\frac{d}{d x} (3 x^{-2})

Apply the differentiation rule:

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

with f =

x

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

\frac{d}{d x} (3 x^{-2})

3 \cdot \frac{d}{d x} (x^{-2})

■The derivative of x^n is n*x^(n-1):

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

-2 x^{-3}

3 \cdot (-2 x^{-3})

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

3 \cdot (-2)

-6

-6 x^{-3}

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

- 6 x^{-3}

- \frac{6}{x^{3}}

- \frac{6}{x^{3}}

Compute f(

1

)

\frac{3}{1^{2}}

3

The point of tangency is (

1

3

)

Compute f'(

1

)

- \frac{6}{1^{3}}

-6

The slope is f'(

1

) =

-6

The tangent passes through this point with this slope:

y = - 6 \cdot (x - 1) + 3

Compute the y-intercept:

b = 3 + 6

3 + 6

ⓘ Start from

3

and move forward

6

on the number line:

3 + 6

9

9

▶Tangent line:

y = - 6 x + 9