Answer
The equation of the tangent line to f(x) = (x - 4)⁴ at x = 5 is given as
y - 1 = 4 (x - 5)
y = 4x - 19
Step-by-step explanation
The steps to finding the equation of the tangent line include
Step 1
Finding the first derivative of f(x).
Step 2
Put x value of the indicated point into the first derivative, f '(x) to find the slope at x.
Step 3
Put x value into f(x) to find the y coordinate of the tangent point.
Step 4
Combine the slope from step 2 and point from step 3 using the point-slope formula to find the equation for the tangent line.
The general form of the equation in point-slope form is
y - y₁ = m (x - x₁)
where
y = y-coordinate of a point on the line.
y₁ = This refers to the y-coordinate of a given point on the line
m = slope of the line.
x = x-coordinate of the point on the line whose y-coordinate is y.
x₁ = x-coordinate of the given point on the line
For this question,
f(x) = (x - 4)⁴
Step 1
First derivative = (df/dx) = f'(x) = 4 (x - 4)³
Step 2
We are asked to find the equation of tangent line at x = 5
f'(x) = 4 (x - 4)³
f'(5) = 4 (5 - 4)³ = 4 (1)³ = 4 (1) = 4
Step 3
f(x) = (x - 4)⁴
f(5) = (5 - 4)⁴ = 1⁴ = 1
y₁ = 1
Step 4
m = 4
x₁ = 5
y₁ = 1
The equation of the tangent line at x = x₁ is given as
y - y₁ = m (x - x₁)
y - 1 = 4 (x - 5)
y - 1 = 4x - 20
y = 4x - 20 + 1
y = 4x - 19
Hope this Helps!!!