Final answer:
To find the derivative of f(x) = x + 1/x, we use the limit definition of a derivative, apply the difference quotient, and then take the limit as h approaches zero. This process yields the derivative f'(x) = 1 - 1/x^2.
Step-by-step explanation:
The student is asking how to find the derivative f'(x) of the function f(x) = x + 1/x using the definition of a derivative.
To find the derivative f'(x), we use the limit definition of the derivative: f'(x) = lim_(h->0) ((f(x+h) - f(x))/h).
First, find f(x+h):
f(x+h) = (x+h) + 1/(x+h)
Then, set up the limit expression:
f'(x) = lim_(h->0) ((x+h + 1/(x+h) - (x + 1/x)) / h)
Simplify the expression inside the limit:
f'(x) = lim_(h->0) ((h - (1/x - 1/(x+h))) / h)
Combine the fraction:
f'(x) = lim_(h->0) ((h - h/((x)(x+h))) / h)
Clean up the expression:
f'(x) = lim_(h->0) (1 - 1/((x^2+xh)))
Finally, take the limit as h approaches 0 to find f'(x):
f'(x) = 1 - 1/x^2