Question

a) Find the derivative of the function using the definition of derivative.
f(b) = b + √b

I know how to find this by differentiation, but I can't seem to find it with the f'(x) = lim (h-->0) [f(x + h) - f(x)] / h formula. Please help me to find this with the definition of derivative.

answer is: 1+0.5b^(-0.5)

Thanks!

Answer 1

(b) = b + sqrt(b)
f(b + h) = b + h + sqrt(b + h)


(b + h + sqrt(b + h) - b - sqrt(b)) / h =>
(b + h - b) / h + (sqrt(b + h) - sqrt(b)) / h =>
h/h + (sqrt(b + h) - sqrt(b)) / h =>
1 + (sqrt(b + h) - sqrt(b)) / h =>
1 + (sqrt(b + h) - sqrt(b)) * (sqrt(b + h) + sqrt(b)) / (h * (sqrt(b + h) + sqrt(b))) =>
1 + (b + h - b) / (h * (sqrt(b + h) + sqrt(b))) =>
1 + h / (h * (sqrt(b + h) + sqrt(b))) =>
1 + 1 / (sqrt(b) + sqrt(b + h))

h goes to 0

1 + 1 / (sqrt(b) + sqrt(b)) =>
1 + 1 / (2 * sqrt(b)) =>
1 + (1/2) * b^(-1/2)

Answer 2

Thank you so much for taking the time to explain! :) Very much appreciated. @Rachel98