Use the definition of the derivative to find the f′(x) for f(x)=x+1x.

Question

Note: Using the definition means doing a limit and algebraically solving the limit. If you use the rules from the next section to do this derivative you will receive NO CREDIT.

he states this under the question any help will be greatly appreciated

Answer 1

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

Simplify the numerator:

[(x+h) + 1/(x+h) - x - 1/x]

[h + 1/(x+h) - 1/x]

Now simplify the complex fraction by multiplying numerator and denominator by x(x+h):

[h + 1/(x+h) - 1/x] / h [x(x+h) / x(x+h)]

[hx(x+h) + x - (x+h)] / [hx(x+h)]

[x²h +h²x + x - x - h] / [hx(x+h)]

h[x² + hx - 1] / [hx(x+h)]

Cancel the factor of h (h approaches zero, but does not equal zero):

[x² + hx -1] / [x(x+h)]

Finally take the limit as h-> 0

lim_h->0 [x² +xh - 1] / [x(x+h)] = [x² + 0 - 1] / [x(x + 0)] = (x² - 1) / x²

This graph shows f(x), f'(x) and g(x) equals the derivative of f(x) calculated above.

It also shows tangent lines at a movable point on the graph of f, using the derivative to determine the slope at that point:

desmos.com/calculator/34jgi2ewnv