g(t) = 7 / sqrt t
by definition
g'(t) =lim_(h to 0) (g(t + h) - g(t))/(h)
=lim_(h to 0) 1/h * (7 / sqrt (t + h) - 7 / sqrt t)
lifting the constant term out
= 7 lim_(h to 0) 1/h * (1 / sqrt (t + h) - 1 / sqrt t)
combining the fractions
=7 lim_(h to 0) 1/h * (sqrt (t ) - sqrt (t+h))/(sqrt (t+h)sqrt (t))
=7 lim_(h to 0) 1/h * (sqrt (t ) - color(red)(sqrt(t)) sqrt (color(blue)(1+h/t)))/(sqrt (t+h)sqrt (t))
cancelling sqrt t
=7 lim_(h to 0) 1/h * (1 - sqrt (1+h/t))/(sqrt (t+h))
Now, by generalised Binomial Theorem:
sqrt (1+h/t) = (1+h/t) ^(1/2) = 1 + 1/2 h/t + O(h^2)
implies 7 lim_(h to 0) 1/h * (1 - (1 + 1/2 h/t + O(h^2)))/(sqrt (t+h))
=7 lim_(h to 0) 1/h * ( - 1/2 h/t + O(h^2))/(sqrt (t+h))
=7 lim_(h to 0) ( - 1/(2t) + O(h))/(sqrt (t+h))
the limit of the quotient is the quotient of the limits where the limits are known
=7 (lim_(h to 0) - 1/(2t) + O(h))/(lim_(h to 0) sqrt (t+h))
=7 ( - 1/(2t) )/( sqrt (t))
=7/(2 sqrt (t^3))
