How do you find the slope of the curve #f(x)=sqrt(x-1)# at the point x=5?
The slope is
(there are two possible values as the function is not one-one as it contains a square root)
The slope of a function at some particular value of the independent variable is the first derivative evaluated at that particular value.
Denoting the two component functions as
it might be noted that
The chain rule states
#= g'(h(x))h'(x) #
That is, the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
It is convenient to write
so that the rules of polynomial differentiation might be easily applied
evaluating this at the value of the inner function
The overall derivative is