Calculus, Derivative By Definition

Question

Locate the derivative of the function  f(x)=( x²+1)/x applying the definition of derivative only.  Then verify your result using the quotient rule of differentiation and the formula for the derivative of power functions.

Answer 1

Hi Jose, Here is a video answer to your revised question. I hope this helps!

Answer 2

Derivative definition:

f(x) = (x² + 1)/x

f(x + h) = ((x + h)² + 1)/(x + h) = (x² + 2xh + h² + 1)/(x + h)

f(x + h) - f(x) = (x² + 2xh + h² + 1)/(x + h) - (x² + 1)/x

Get a common denominator:

[x(x² + 2xh + h² + 1)]/[x(x + h)] - [(x + h)(x² + 1)]/[x(x + h)]

(x³ + 2x²h + xh² + x)/[x(x + h)] - (x³ + x²h + x + h)/[x(x + h)]

Combine numerators:

(x³ - x³ + 2x²h - x²h + xh² + x - x - h)/[x(x + h)]

(x²h + xh² - h)/[x(x + h)]

[h(x² + xh - 1)]/[x(x + h)]

Finally, [f(x + h) - f(x)]/h = [h(x² + xh - 1)]/[x(x + h)]/h = (x² + xh - 1)/[x(x + h)]

The limit of this as h → 0 = (x² - 1)/x²

Using the quotient rule (where f(x) = u/v, then f '(x) = u'v - uv')/v²)

u = x² + 1 so u' = 2x (power rule)

v = x so v' = 1 and v² = x²

So f '(x) = [2x(x) - (x² + 1)(1)]/x² = (2x² - x² - 1)/x² = (x² - 1)/x²