Derivation Problem: Struggling with u Function

[Gekkoo]

Homework Statement

I don't manage to derive a function properly :(

Homework Equations

u = x^α * [(m-Ax)/B]^β

The Attempt at a Solution

FOC: x^α*ln(x)*?=0

It is the second factor I have a problem with. Preciate any help!

 

[CompuChip]

Are you sure this is precalculus? Because it is not nearly as simple as you seem to think it should be.
Let me assume that your variable is x and that all other letters represent constants.

Then first of all, there are two functions of x being multiplied, so you will need the product rule:
(x^α * [(m-Ax)/B]^β)' = (x^α)' * [(m-Ax)/B]^β + x^α * ( [(m-Ax)/B]^β )'
Then the derivative of x^α is not x^α ln(x), but if x is the variable it's simply α x^{α - 1}. For the derivative of the second part, you will need the chain rule (you can write it as y^β and get β y^{β - 1} dy/dx).If, for some strange reason, you want to take the derivative with respect to α, however, you are right: you simply get
x^α ln(x) * [(m-Ax)/B]^β
because the whole second factor is simply constant with respect to α.

 

[HallsofIvy]

The natural logarithm only turns up in derivatives of exponential problems, such as the derivative of a^x- that is, with the variable, x, in the exponent.

Problems like this, which is just a power of x, with x in the base, are done by the 'power law', (x^a)'= ax^{a- 1} which is true for a and number, not just integers.

 

[berkeman]

(thread moved to Calculus & Beyond)