What is the Proof of the derivative of the third root of x?

Answer 1

You can easily derive it from formula for the derivative of a power, if you remember that the cubic root of x is equal to x1/3.

This question asks for the proof of the derivative, not the derivative itself.

Using the definition of derivative, lim f(x) as h approaches 0 where f(x) = (f(a+h)-f(a))/h, we get the following:

[(a+h)1/3 - a1/3]/h

Complete the cube with (a2 + ab + b2)

Multiply by [(a+h)2/3 + (a+h)1/3 × a1/3 + a2/3] / [(a+h)2/3 + (a+h)1/3 × a1/3 + a2/3]

This completes the cube in the numerator, resulting in the following:

(a + h - a) / (h × [(a+h)2/3 + (a+h)1/3 × a1/3 + a2/3])

h / (h × [(a+h)2/3 + (a+h)1/3 × a1/3 + a2/3])

h cancels

1 / [(a+h)2/3 + (a+h)1/3 × a1/3 + a2/3]

Now that we have a function that is continuous for all h, we can evaluate the limit by plugging in 0 for h.

This gives

1/[a2/3 + a1/3 × a1/3 + a2/3]

Simplify a1/3 × a1/3

1/[a2/3 + a2/3 + a2/3]

(1/3)a2/3 or (1/3)a-2/3

This agrees with the Power Rule.