Find the rate of change of the reciprocal function 𝑓(𝑥)=1/𝑥 with respect to 𝑥 when 𝑥=2, 3, and 4. (Use symbolic notation and fractions where needed.)

Question

𝑓′(2)=

𝑓′(3)=

𝑓′(4)=

Answer 1

I'm not sure but I'm going to guess that you are using the limit definition of the derivative to find these.

The limit definition is:

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

But to find f(x+h) - f(x) or 1/(x+h) - 1/x we must have a common denominator:

Multiply 1/(x+h) by x/x and multiply 1/x by (x+h)/(x+h) so:

f(x+h) - f(x) = x/[x(x+h)] - (x+h)/[x(x+h)]

f(x+h) - f(x) = (x - (x+h))/[x(x+h)]

f(x+h) - f(x) = (x - x - h)/[x(x+h)]

f(x+h) - f(x) = (-h)/[x(x+h)]

f(x+h) - f(x) = (-h)/(x² + xh)

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

And the limit of this as h approaches zero is -1/x²

So f '(x) = -1/x²

Therefore f '(2) = -1/(2)² = -1/4

And f '(3) = -1/(3)² = -1/9

And f '(4) = -1/(4)² = -1/16