For a fixed positive constant

MATHEMATICS · Multi-part · CBSE · Mathematics · Limits and Derivatives · Limits and Derivatives

Problem

a

, consider the function

h(x)=\sqrt{x+a}-\sqrt{a}

defined for

x>-a

. (a) Express the difference quotient

\dfrac{h(x+h)-h(x)}{h}

, where

h\neq 0

. Rewrite the numerator so that no radicals appear in it. (b) Evaluate

\displaystyle \lim_{h\to 0}\dfrac{h(x+h)-h(x)}{h}

to obtain an explicit formula for

h'(x)

valid on the domain of

h

. (c) Determine all real numbers

x

for which

h'(x)

exists. Support your answer by referencing both the domain of

h

and the steps in part (b). (d) Suppose

a=9

. Calculate

h'(7)

and interpret this value as an instantaneous rate of change of

h

with respect to

x