Question

For the following exercises, use the definition of a derivative to find f′(x).
f(x) = x+1/x

Answer 1

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