Question

Let z = x^20/y^10 where x = e^5t and y= sin 2t. Use the chain rule to find dz/dot when t = pi/4

Answer 1

To find dz/dt, we use the chain rule to find the derivatives of x and y with respect to t, and then substitute these values into the equation for dz/dt. Finally, we can calculate the value of dz/dt at t = pi/4.

Step-by-step explanation:

To find dz/dt using the chain rule, we need to find the derivatives of x and y with respect to t and then substitute these values into the given equation for dz/dt.

We have x = e^(5t), so dx/dt = 5e^(5t).

We have y = sin(2t), so dy/dt = 2cos(2t).

Substituting these derivatives into the equation dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt), we have dz/dt = 20(t^19/e^10t)(5e^(5t)) - 10(t^9e^10t/sin^11(2t))(2cos(2t)).

Substituting t = pi/4, we can calculate the value of dz/dt.