Question #27dca

2 Answers
Mar 22, 2016

See below

Explanation:

a. Given f(x)=x
To find (f(x+h)-f(x))/h ......(1)

Clearly f(x+h)=x+h
Inserting in equation (1), we obtain
((cancel x+h)-(cancel x))/h
h/h=1, for all values of h except h=0 or oo where division is not defined as well as oo/oo is indeterminate.
b. lim (f(x+h)-f(x))/h-=f'(x)
color(white){W}h->0

In a. we have already shown that LHS=1 so long as h!=0
Therefore lim (f(x+h)-f(x))/h=f'(x)=1
color(white){WWWW}h->0
c. From b. f'(x)=1
Now f'(2) means first derivative of function f(x) at x=2.
Since first derivative has been found to be constant=1, therefore it is independent of the value of x.
Hence, f'(2)=1.

Mar 22, 2016

To think of it as a power function, recall that x = x^1

Explanation:

Applying the power rule, we get

f'(x) = 1x^(1-1)

= 1x^0

But x^0 = 1, so we finish with

f'(x) = 1