find the derivative of f(x)=1/x+3 at x=3 using the derivative defintion

Question

Using the definition of a derivative

Answer 1

Given function f(x) = 1/x+3

definnition of derivatives

Let y = f(x) be  a function ,the derivative of f is function whose value at x is the limit

f'(x) = lim(h-->0) [f(x+h)-f(x)]/h

f(x+h) = 1/(x+h+3)

f(x) = 1/x+3

f'(x) = lim(h-->0)[1/(x+h+3)-1/x+3]/h

= lim(h-->0) [(x+3-x-h-3)/(x+h+3)(x+3)]/h

= lim(h-->0) [-h/(x+h+3)(x+3)]/h

= lim(h-->0)-h/h(x+h+3)(x+3)

= lim(h-->0) -1/ (x+h+3)(x+3)

= -1/(x+3)^2

at x = 3

= -1/(3+3)^2

= -1/36