Evaluate the derivatives by implicit differentiation. Assume all letters represent constants, except for the independent and dependent variables occurring in the derivative.
𝑃𝑉^𝑐=𝑛𝑅𝑇
-c*P/V good
Assuming that only c and n are constants, then we have
V^c dP/dT + cPV^(c-1) dV/dT = n(T dR/dT + R dT/dT)
you can replace /dT with any other varying quantity, such as time, etc.
Somehow I don't think this is what you are looking for, but since your question was rather vague, it's the best I can do so far.
All you really need for implicit derivatives is the chain rule, product rule, quotient rule.
Hi oobleck, the question asks for DP/DV, sorry for not including this earlier.
wrong
To evaluate the derivatives by implicit differentiation, we need to follow these steps:
1. Identify the dependent variable: In this case, the dependent variable is 𝑃𝑉.
2. Identify the independent variable: In this case, the independent variable is not explicitly given. However, we can assume it to be 𝑇.
3. Take the derivative of both sides of the equation with respect to the independent variable:
For the left-hand side (LHS), 𝑃𝑉^𝑐, we can apply the power rule. The derivative of 𝑢^𝑣, where 𝑢 is a function of 𝑥, is given by 𝑣𝑢^(𝑣−1)𝑢'.
Therefore, the derivative of 𝑃𝑉^𝑐 with respect to 𝑇 is 𝑐𝑃𝑉^(𝑐−1)𝑃𝑉'.
For the right-hand side (RHS), 𝑛𝑅𝑇, we have a product of three terms 𝑛, 𝑅, and 𝑇. The derivative of a product can be calculated by applying the product rule. The product rule states that if we have two functions 𝑢(𝑇) and 𝑣(𝑇) and want to find the derivative of their product, then the derivative is given by 𝑢𝑣' + 𝑣𝑢'.
Applying the product rule, the derivative of 𝑛𝑅𝑇 with respect to 𝑇 is 𝑛𝑅 + 𝑅𝑇'.
4. Set the derivatives equal to each other and solve for the unknown derivative:
Equating the derivatives obtained in step 3, we have 𝑐𝑃𝑉^(𝑐−1)𝑃𝑉' = 𝑛𝑅 + 𝑅𝑇'.
Now, solve for 𝑃𝑉' by rearranging the terms:
Divide both sides by 𝑐𝑃𝑉^(𝑐−1):
𝑃𝑉' = (𝑛𝑅 + 𝑅𝑇') / (𝑐𝑃𝑉^(𝑐−1))
Therefore, the derivative of 𝑃𝑉 with respect to 𝑇 is given by (𝑛𝑅 + 𝑅𝑇') / (𝑐𝑃𝑉^(𝑐−1)).