# A = 1/2 ab Sinè

How is dA/dt related to da/dt, db/dt and dè/dt

dA/dt= 1/2 b SinTheta da/dt + 1/2 a SinTheta db/dt + 1/2 ab cosTheta dTheta/dt

This is a U = xyz form

dU/dt= xy dz/dt + ...

## To understand how dA/dt is related to da/dt, db/dt, and dè/dt, we need to break down the expression for dA/dt.

Given A = (1/2)abSinè, where a, b, and è are variables, the derivative of A with respect to t (dA/dt) can be calculated using the product rule of differentiation. The product rule states that the derivative of a product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.

Applying the product rule to A = (1/2)abSinè, we get:

dA/dt = (1/2)(bSinè)(da/dt) + (1/2)(aSinè)(db/dt) + (1/2)(ab)(cosè)(dè/dt)

So, dA/dt is equal to three terms:

Term 1: (1/2)(bSinè)(da/dt) - This represents the change in A due to the change in variable a, with b and è being constant. We multiply bSinè by da/dt to account for the effect of the change in a on A.

Term 2: (1/2)(aSinè)(db/dt) - This represents the change in A due to the change in variable b, with a and è being constant. We multiply aSinè by db/dt to account for the effect of the change in b on A.

Term 3: (1/2)(ab)(cosè)(dè/dt) - This represents the change in A due to the change in variable è, with a and b being constant. We multiply abcosè by dè/dt to account for the effect of the change in è on A.

So, dA/dt is the sum of these three terms, each representing the effect of the respective variables (a, b, and è) on the rate of change of A with respect to time (t).

It's important to note that the expression you mentioned, U = xyz, and dU/dt = xy(dz/dt) + ..., is not directly related to the question you posed regarding dA/dt and its relation to da/dt, db/dt, and dè/dt.