Let f(t) = cubic root of t. For a ≠ 0, find f ′(a).

Answer 1

Another way to express a root, is with a fractional exponent, like so:

square root of x = x^1/2

cube root of x = x^1/3

So, we can write the problem in the following way,

Let f(t) = t^1/3

Taking the derivative by using the power rule (multiply by the exponent, then subtract 1 from the exponent)

f '(t) = (1/3)t^-2/3

Now we can see why the problem states that a ≠ 0. If a was allowed to be 0, then we would have a 0 in the denominator.

f '(a) = (1/3)a^-2/3 = 1 / (3a^2/3)