The discussion focuses on calculating the slope of the curve defined by the function f(x) = x^(1/3) at the point x = 1 using the definition of the derivative. The limit expression f'(x) = lim (x->1) [f(x) - f(1)] / (x - 1) is established, where f(1) = 1. Participants emphasize the importance of correctly applying the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2) to rationalize the numerator in the limit calculation.
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Kazane said:Homework Statement
f(x)=x1/3
use the definition of f'(a) to calculate the slope of the curve at x=1
(Hint: by rationalizing the numerator. useful formula a3 -b3=(a-b)(a2+ab+b2)
The Attempt at a Solution
f'(x)=lim (x->1) f(x)-f(1)/x-1
=lim (x->1) x1/3-11/3/ x-1
How can I do x1/3-11/3 to a3 -b3=(a-b)(a2+ab+b2)?
I'm confused so please help!
Let a= x^{1/3}, b= y^{1/3}Kazane said:Homework Statement
f(x)=x1/3
use the definition of f'(a) to calculate the slope of the curve at x=1
(Hint: by rationalizing the numerator. useful formula a3 -b3=(a-b)(a2+ab+b2)
The Attempt at a Solution
f'(x)=lim (x->1) f(x)-f(1)/x-1
=lim (x->1) x1/3-11/3/ x-1
How can I do x1/3-11/3 to a3 -b3=(a-b)(a2+ab+b2)?
I'm confused so please help!