Consider the function, f(x) = √x-4 (square root goes over x-4) at the point (8,2) find the slope of the tangent line to the curve using the definition of the derivative

Question

And find the equation of the tangent line. I am confused on how to treat the square root as I haven’t covered a question using them like this yet.

Answer 1

lim h>0

[f(8+h)-f(8)]/ h = [√(4+h) - 2] / h

Multiply top and bottom by conjugate of √ (4+h) - 2 which is √(4+h) + 2

√[4+h) - 2 √(4+h) + 2

------------ -------------

h √(4+h) +2

4+h - 4

----------

h [ √(4+h) + 2]

h

------

h [ √4+h + 2]

1

------

√ [4+h] + 2

As h > 0 this > (1/4).

So derivative is 1/4.