What is the second derivative of f(x)= x^(2/3) (6-x)^(1/3)?

[FChebli]

Homework Statement

Find the second derivative of the function:

f(x)= x^(2/3) (6-x)^(1/3)

Homework Equations

The chain, product and quotient rules

The Attempt at a Solution

I have found the first derivative and checked my solution:

′()= 4− / ^(1/3) (6−)^(2/3)

The final solution is supposed to be:

''()= -8 / ^(4/3) (6−)^(5/3)

I know almost all the steps but I couldn't reach the final answer! Can you please help me?
Thanks in advance!

 

[Mark44]

Homework Statement

Find the second derivative of the function:

f(x)= x^(2/3) (6-x)^(1/3)

Homework Equations

The chain, product and quotient rules

The Attempt at a Solution

I have found the first derivative and checked my solution:

′()= 4− / ^(1/3) (6−)^(2/3)

There's a difference between what you think you wrote and what you actually wrote.

Here's how knowledgeable people would interpret what you wrote:
$$f'(x) = 4 - \frac{x}{x^{1/3}}(6 - x)^{2/3}$$

Since you undoubtedly meant for 4 - x to appear in the numerator, you need more parentheses or brackets, in both top and bottom, like so:
f'(x) = (4 - x)/[x^1/3(6 - x)^2/3]

Since you need to take the derivative again, it might be easier to leave the derivative as got it the first time; i.e., as a product with negative exponents. It looks nicer by changing the negative exponents to positive exponents, but taking the derivative this time means using the quotient rule. If you leave the first derivative as a product, you can use the product rule, which is a bit simpler than the quotient rule, hence less prone to errors.

The final solution is supposed to be:

''()= -8 / ^(4/3) (6−)^(5/3)

I know almost all the steps but I couldn't reach the final answer! Can you please help me?
Thanks in advance!