compute f'(c) for each of the following using the alternate from of the definition of the derivative

Question

compute f'(c) for each of the following using the alternate from of the definition of the derivative

f(x)=3x+1, c=1

f(x)=x+4/x. c=4

show works

Answer 1

The alternative definition of the derivative is as follows:

the derivative of f at c

f'(c) = lim ((f(x)-f(c)) / (x-c))

x -> c

the two functions given in the problem are

1. f(x)=3x+1, c=1
2. f(x)=x+(4/x), c=4

Now let's try

#1 f(x)=3x+1, c=1

f(1) = 3(1) + 1 = 3+1 = 4

f'(1) = lim ((3x-1) - 4) / (x-1)

x->1

simplify --> lim (3x-3)/(x-1) = lim (3(x-1))/(x-1) = 3

(x-1) term cancels out and we have f'(1) = 3

#2 f(x)=x+(4/x), c=4

f(4) = 4 + (4/4) =4+1 = 5

f'(4) = lim ((x+(4/x)) - 5) / (x-4)

x->4

simplify the numerator first: x+(4/x)) - 5 , multiply the first and last term by x/x

(x^2)/x + (4/x) - (5x/x) => ((x^2) -5x + 4) / x => ((x-1)(x-4))/x

simplify --> lim [(x-1)(x-4)] / [(x-4)x] , (x-4) cancels out. we have

lim (x-1)/x as x --> 4, plug four into x we have 3/4

f'(4) = 3/4

you can check your answers by using the standard derivative form.