How do you find the derivative using limits of #f(x)=1/x^2#?

Question

Christopher Allers · Accepted Answer

#lim_(Deltax->0) (1/(x+Deltax)^2-1/x^2)/(Deltax) = -2/x^3#

The limit definition of the derivative of a function #f(x)# is:

#f'(x) = lim_(Deltax->0) (f(x+Deltax)-f(x))/(Deltax) = lim_(Deltax->0) (Deltaf)/(Deltax)#

Let's calculate the increment of the function between #x# and #x+Deltax#:

#Delta f = 1/(x+Deltax)^2 -1/x^2 = (x^2 - (x+Deltax)^2)/(x^2(x+Deltax)^2) = (x^2 - x^2 -2xDeltax -(Deltax)^2)/(x^2(x+Deltax)^2) = (-2xDeltax -(Deltax)^2)/(x^2(x+Deltax)^2)#

The incremental ratio is then:

#(Deltaf)/(Deltax) =(-2x -Deltax)/(x^2(x+Deltax)^2)#

and passing to the limit:

#lim_(Deltax->0) (-2x -Deltax)/(x^2(x+Deltax)^2) = -2x/(x^2*x^2) = -2/x^3#

Samantha Adkison · Answer

#f'(x)=color(green)(-2 * 1/x^3)#
#color(white)("XXXXXX")#(see below for determination using limits)

Using the limit definition for a derivative: #color(white)("XXX")f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h# For the case #f(x)=1/x^2# #f(x+h)-f(x) = 1/(x+h)^2-1/x^2# #color(white)("XXXXXXXXXX")=(x^2 - (x+h)^2)/(x^2 * (x+h)^2)# #color(white)("XXXXXXXXXX")=(-2xh-h^2)/(x^4+2hx^3+h^2x^2)# and theerfore #(f(x+h)-f(x))/h = (-2x-h)/(x^4+2hx^3+h^2x^2)# This is defined when #h=0# so #lim_(hrarr0) (f(x+h)-f(x))/h=(-2x-0)/(x^4+2 * 0 * x^3 + 0^2 * x^2) # #color(white)("XXXXXXXXXXXXX")=(-2x)/(x^4) = -2/x^3=(-2) * (1/x^3)#

Evelyn Agard · Answer

undefined To find the derivative of $f(x) = \frac{1}{x^2}$, apply the definition of the derivative, which involves taking the limit of the difference quotient as $h$ approaches 0.