Let f(x) be the function given below. If a isnt 0, use the following formula for the derivative to find f '(a)

Question

f(x)=cubed-root of x
f(a)=limit x goin' to a... f(x)-f(a)/x-a
please help.. i got 1 over 3 times (the squared-root of a)^2
oooh no!! i have no idea! someone save me!!

Answer 1

f(x) = x^(1/3)

[f(x) - f(a)]/(x - a) = [x^(1/3) - a^(1/3)/(x - a) =
= [x^(1/3) - a^(1/3)/{[x^(1/3)]^3 - [a^(1/3)]^3} =
= [x^(1/3) - a^(1/3)]/{[x^(1/3)-a^(1/3)][x^(2/3) +
x^(1/3) * a^(1/3) + a^(2/3)] =
= 1/[x^(2/3) + x^(1/3)*a^(1/3) + a^(2/3)]

Now f'(a) = lim (x->a) 1/[x^(2/3) + x^(1/3)*a^(1/3) + a^(2/3)] =
= 1/[a^(2/3) + a^(2/3) + a^(2/3)] = 1/[3a^(2/3)]


Or 1 over 3 times (the cusbed-root of a)^2

Answer 2

f(x) = x^(1/3)
Your formula tesults in 0/0, an indeterminate form.
To find the limit we must use L/Hospital's rule which requires us to take the derivatives of the numerator and denominator separately. We get (1/3)x^-(2/3)/1 = 1/3x^2/3.

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

Why not use the power rule to begin with? Or haven't you got there yet?