Question

5.2.) Find the slope of the tangent line

Answer 1

To derive the function, we will use the power rule.

Power rule is expressed with the following formula:


`(d)/(dx) x^n = n \cdot x^(n-1)`

Use this rule to derive both terms in the function:


`(d)/(dx) [6x - x^2] = (d)/(dx) 6x - (d)/(dx) x^2`


`(d)/(dx) 6x = 6`

`(d)/(dx) x^2 = 2x`


`(d)/(dx) [6x - x^2] = 6 - 2x`

We can now plug in the x-value for this derivative to find the slope of the tangent line at said x-value:


`f'(4) = 6 - 2(4) = 6 - 8 = \boxed{-2}`

The slope of the tangent line at x = 4 will be -2.

Answer 2

Use the power rule to find the derivative of f(x). That rule is
d(x^n)/dx = n·x^(n-1)

f'(x) = 6 -2x
Then at x=4 ...
f'(4) = 6 -2·4 = -2