Find the derivative: f(x)=1/cubed root(3-x^3)
A. -1/3(3-x^3)^4/3
B. x^2/(3-x^3)^4/3
C. -x^2/(3-x^3)^2/3
D. -x^2/(3-x^3)^4/3
E. None of these
y = (3 -x^3)-(1/3)
dy/dx = -(1/3) (3-x^3)^-2/3 (-3x^2)
= + x^2 /(3-x^3)^(2/3)
E
Hmmm. -1/3 - 1 = -4/3
I get (B)
My arithmetic was wrong, go with Steve !
To find the derivative of the function f(x)=1/cubed root(3-x^3), we can use the chain rule. The chain rule states that if we have a composite function, we can find its derivative by taking the derivative of the outer function and multiplying it with the derivative of the inner function.
Let's start by rewriting the function in a different form to make it easier to work with. Recall that the cubed root of a number can also be written as the number raised to the power of 1/3. Therefore, f(x) can be rewritten as:
f(x) = (3-x^3)^(-1/3)
Now, let's find the derivative using the chain rule. The outer function is the power of -1/3, and the inner function is the expression (3-x^3). The derivative of the outer function is obtained by multiplying the derivative of -1/3 with the original function raised to the power of -1/3 - 1. The derivative of the inner function can be found by applying the power rule.
Using the chain rule, the derivative of f(x) is:
f'(x) = (-1/3)(3-x^3)^(-1/3 - 1) * (d/dx)(3-x^3)
Now, let's find the derivative of the inner function, which is (d/dx)(3-x^3). Applying the power rule, we get:
(d/dx)(3-x^3) = -3x^2
Substituting this into the derivative expression, we have:
f'(x) = (-1/3)(3-x^3)^(-1/3 - 1) * (-3x^2)
Simplifying further, we have:
f'(x) = (x^2)(3-x^3)^(-4/3)
Therefore, the correct option is B. x^2/(3-x^3)^(4/3).