Understanding a function $A(W+r)^{b}$.

Question

I am given the following function $$Q = A(W+r)^{a},$$ where $A$ and $r$ are constants. I am asked to do several things with this function:

1. Find $\frac{\partial Q}{\partial W}$.
2. Given constants $a$, $A$, $b$, $B$ and $r$, solve the following equation for $W$, $$A(W+r)^a = B(W+r)^b.$$

I am unfamiliar with how to expand this type of function, and also find it difficult to differentiate it in its current form.

Is it possible to expand $Q$? And if not, what approach do I take to the two uses of it?

Answer 1

You do not need to expand the expression to do any of the things you're asked. If you're having trouble understanding how to do these things with all the variables around, think about how you'd do them if there were only one variable.

For example, if we choose values for $A$, $a$ and $r$, we might get, for example:

$Q = 100(W + 5)^3$

Can you find $\frac{dQ}{dW}$?

If we had:

$100(W + 5)^3 = 75(W + 5)^4$

could you find $W$?