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Hello, please help me, I'm studying for a test. :)

I won't say the question because I'd like to figure it out myself or try first, but I'm looking for a simpler way to go about this.

I have to find the derivative of h(x)= (x)(x-1)^1/3

I got: h'(x)= (x-1)^1/3 + (1/3)(x)(x-1)^-2/3

I'm looking for a way to combine the two terms, but the cube roots are messing me up and I'm not completely sure how to go about it... been stuck on this for about an hour...

If any one can help me simplify h'(x) and, if you'd like, show me how, that would be amazing!

Thank you for your time.

Not much joy here. You could always factor out the (x-1)^-2/3 to get

(x-1)^-2/3 * (x-1 + x/3)
= (x-1)^-2/3 * (4x/3 -1)

Depending on what comes next, you may want to massage it some.

:D I'm so happy! Never thought to factor that out x) thank you!

Of course, I'd be happy to help you simplify the derivative of h(x)= (x)(x-1)^(1/3).

To combine the two terms in h'(x), we need to find a common denominator for them.

First, let's write the terms with a common denominator of 3(x-1)^(2/3):

h'(x) = (x-1)^(1/3) + (1/3)(x)(x-1)^(-2/3)

To combine these terms, let's multiply the first term by 3(x-1)^(2/3) and the second term by (x-1)^(1/3):

h'(x) = 3(x-1)^(2/3) * (x-1)^(1/3) + (1/3)(x)(x-1)^(-2/3)

Now, we can simplify each term separately:

For the first term, when we multiply (x-1)^(2/3) and (x-1)^(1/3), we need to add the exponents:

(x-1)^(2/3) * (x-1)^(1/3) = (x-1)^((2/3)+(1/3)) = (x-1)^(3/3) = (x-1)

So, the first term simplifies to (x-1).

For the second term, we have (1/3)(x)(x-1)^(-2/3). Let's simplify the denominator first by using the fact that (x-1)^(-1/3) = 1/(x-1)^(1/3):

(x-1)^(-2/3) = [(x-1)^(-1/3)]^2 = (1/(x-1)^(1/3))^2 = 1/(x-1)^(2/3)

Now, we can rewrite the second term with the simplified denominator:

(1/3)(x)(x-1)^(-2/3) = (1/3)(x)/[1/(x-1)^(2/3)] = (1/3)(x)(x-1)^(2/3)

Therefore, the second term simplifies to (1/3)(x)(x-1)^(2/3).

Now, let's combine the simplified terms:

h'(x) = (x-1) + (1/3)(x)(x-1)^(2/3)

And that's the simplified form of the derivative of h(x).