Question

Calculate the partial derivative dtdu and dudt using implicit differentiation of tu−v.

A) dtdu=u−v,dudt=u−v1
B) dtdu=1,dudt=vu
C) dtdu=t,dudt=t1
D) dtdu=t,dudt=t

Answer 1

To find dtdu and dudt using implicit differentiation of tu-v, we differentiate both sides of the equation with respect to u and v separately. The partial derivative dtdu is equal to t, and the partial derivative dudt is equal to u-v.

Step-by-step explanation:

In order to find the partial derivatives dtdu and dudt of the function tu-v using implicit differentiation, we treat t as a function of both u and v. We differentiate both sides of the equation with respect to u, treating v as a constant, to find dtdu. Similarly, we differentiate both sides of the equation with respect to v, treating u as a constant, to find dudt.

To find dtdu:

dtdu = d(tu-v)/du

Using the power rule, we can differentiate tu-v with respect to u:

dtdu = t(du/d) + u(dt/du) - 0

Since du/d = 1, the term t(du/d) becomes t. And since dt/du = 0, the term u(dt/du) becomes 0. Therefore, dtdu = t.

To find dudt:

dudt = d(tu-v)/dt

Using the power rule, we can differentiate tu-v with respect to t:

dudt = u(dt/d) - v

Since dt/d = 1, the term u(dt/d) becomes u. Therefore, dudt = u - v.