Use the Limit Definition of the Derivative to solve the following:

Question

ℎ(𝑥) = 𝑥³ + 𝑥² + 1 , find ℎ ′ (𝑎), where a is a positive constant

Answer 1

By definition, derivative of function f(x) is defined as

f'(x)=Lim_{h->0}f(x+ax)-f(x))/h <------------- (1)

Another,definition is

f'(a)=Lim_{x->a}f(x)-f(a))/(x-a) <------------- (1)

We use eq (2):

h'(a)= lim _x-->a (h(x)-h(a))/(x-a) = lim _x-->a (𝑥³ + 𝑥² + 1 - (a³ + a² + 1 ))/(x-a)=

= lim _x-->a (𝑥³ - a³ +x²- a² )/(x-a)=

= lim _x-->a (𝑥² +ax+ a² +x+a)= 3a² +2a.