differentiating a cost function using chain rule

Question

How to differentiate the function (cost function)

$$C(x)=V\left(\frac{px-W}{q}\right)$$ where $W,p,q$ are constants,and $W$ representing the consumer's budget or wealth.
$V$ is a function of $x$.

Please explain me how to obtain the first and second derivatives of $C(X)$

Answer 1

To differentiate $C(x)$ you have to apply the chain rule.

$$C'(x)= V'\left(\frac{px-W}{q}\right)\cdot\left(\frac{px-W}{q}\right)' = V'\left(\frac{px-W}{q}\right)\cdot\frac{p}{q}$$

For the second derivitive again:

$$C''(x) = \left(C'(x)\right)' = \left(V'\left(\frac{px-W}{q}\right)\cdot\frac{p}{q}\right)' = V''\left(\frac{px-W}{q}\right)\cdot\frac{p^2}{q^2}$$

Answer 2

Depending on v, you can write it as the following:

$$\frac{dV(\eta)}{d\eta}\cdot\frac{d\eta}{dx}$$

Via the chain rule. I have simply set $\frac{px-W}{q}$ to $\eta$. The derivative depends on what V is.