How do I take the derivative in respect to c1 for the following utility function?
u1 = log(c1) + log [1/2(x-c1-c2)]
details please
To take the derivative of the utility function u1 with respect to c1, you can follow these steps:
1. Start with the utility function: u1 = log(c1) + log [1/2(x-c1-c2)].
2. Take the derivative of each term separately using the rules of calculus.
- For the first term, log(c1), its derivative is simply 1/c1.
- For the second term, log [1/2(x-c1-c2)], we will use the chain rule. Let's denote the expression inside the logarithm as g(x) = 1/2(x-c1-c2). The derivative of this term will be [1/g(x)] * [dg(x)/dc1].
3. Calculate the derivative of the second term using the chain rule.
- dg(x)/dc1 will be -(1/2). The negative sign comes from the fact that c1 appears with a negative sign in g(x).
4. Combine the derivatives of the two terms.
The derivative of the utility function u1 with respect to c1 will be:
du1/dc1 = 1/c1 - [1/2]/[1/2(x-c1-c2)]
Simplifying further:
du1/dc1 = 1/c1 - 1/(x-c1-c2)
That's how you can take the derivative of the utility function u1 with respect to c1.