How do you find the derivative of #f(x) = 1/sqrt(2x-1)# by first principles?
2 Answers
derivative of function to a power
Explanation:
You want the derivative of f to the power
Chain rule
The factor 2 comes from the derivative of f itself
Use limit definition of derivative to find:
#f'(x) = -(2x-1)^(-3/2)#
Explanation:
#f(x) = 1/sqrt(2x-1)#
#f'(a) = lim_(h->0) (f(a+h) - f(a))/h#
#=lim_(h->0) (1/sqrt(2a+2h-1) - 1/sqrt(2a-1))/h#
#=lim_(h->0) (sqrt(2a-1)-sqrt(2a+2h-1))/(h sqrt(2a+2h-1) sqrt(2a-1))#
#=lim_(h->0) ((sqrt(2a-1)-sqrt(2a+2h-1))(sqrt(2a-1)+sqrt(2a+2h-1)))/(h sqrt(2a+2h-1) sqrt(2a-1) (sqrt(2a-1)+sqrt(2a+2h-1)))#
#=lim_(h->0) ((2a-1)-(2a+2h-1))/(h sqrt(2a+2h-1) sqrt(2a-1) (sqrt(2a-1)+sqrt(2a+2h-1)))#
#=lim_(h->0) (-2h)/(h sqrt(2a+2h-1) sqrt(2a-1) (sqrt(2a-1)+sqrt(2a+2h-1)))#
#=lim_(h->0) (-2)/(sqrt(2a+2h-1) sqrt(2a-1) (sqrt(2a-1)+sqrt(2a+2h-1)))#
#=(-2)/(sqrt(2a-1) sqrt(2a-1) (sqrt(2a-1)+sqrt(2a-1)))#
#=(-1)/((2a-1)^(3/2))#
#=-(2a-1)^(-3/2)#
So:
#f'(x) = -(2x-1)^(-3/2)#