Find the slope of the line tangent to the curve

Question

Find the slope of the line to the curve at the point P

Use the formula slope of tangent = lim [x -> c] {f(x) - f(c)} / (x-c)

f(x) = √(x 6), P(3,3)

Answer 1

Let s = lim [x -> c] {f(x) - f(c)} / (x-c)

Putting x = c + h:

s = lim (h -> 0) [ { (c + h + 6)^(1 / 2) - (c + 6)^(1 / 2) } / h ]

Multiplying and dividing (c + h + 6)^(1 / 2) - (c + 6)^(1 / 2) by (c + h + 6)^(1 / 2) + (c + 6)^(1 / 2) gives:

[ (c + h + 6) - (c + 6) ] / [ (c + h + 6)^(1 / 2) + (c + 6)^(1 / 2) ]

= h / [ (c + h + 6)^(1 / 2) + (c + 6)^(1 / 2) ]

Therefore:

s = lim (h -> 0) ( h / { h [ (c + h + 6)^(1 / 2) + (c + 6)^(1 / 2) ] } )

= lim (h -> 0) ( 1 / [ (c + h + 6)^(1 / 2) + (c + 6)^(1 / 2) ] )

= 1 / [ (c + 6)^(1 / 2) + (c + 6)^(1 / 2) ]

= 1 / { 2(c + 6)^(1 / 2) }

Putting c = 3 gives:

1 / { 2 sqrt(9) }

= 1 / 6.