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
Find dP/dV
if c , n , R and T are constant then
P c V^(c-1)dV + V ^c dP = 0
P c V^c dV/V + V^c dP = 0
P c dV + V dP = 0
P c + V dP/dV = 0
dP / dV = -P c/ V
Thank you so much Damon, this has gotten me closer to understanding this type of problem however I get a memo:
"V cannot be part of the answer"
oops it was a matter of Case, thank you!
To find the derivative dP/dV using implicit differentiation, we will differentiate both sides of the equation with respect to V.
The given equation is P * V^c = n * R * T.
Step 1: Differentiate both sides of the equation:
d/dV (P * V^c) = d/dV (n * R * T)
Step 2: Apply the product rule and power rule:
dP/dV * V^c + P * c * V^(c-1) = 0
Step 3: Simplify the equation:
dP/dV * V^c = - P * c * V^(c-1)
Step 4: Solve for dP/dV:
dP/dV = (-P * c * V^(c-1)) / (V^c)
Therefore, the derivative dP/dV is given by (-P * c * V^(c-1)) / (V^c).