Question

If z=sin(x/y), x=4t, y=5−t², find dz/dt using the chain rule. Assume the variables are restricted to domains on which the functions are defined.

dz/dt= _______

Answer 1

To find dz/dt using the chain rule for z=sin(x/y) with x=4t and y=5-t², differentiate x and y with respect to t, then apply the chain rule to find the rate of change of z with respect to t.

Step-by-step explanation:

To find the derivative dz/dt using the chain rule for the function z=sin(x/y), where x=4t and y=5-t², we will first need to find dx/dt and dy/dt.

Step 1: Differentiate x with respect to t:
dx/dt = d(4t)/dt = 4.

Step 2: Differentiate y with respect to t:
dy/dt = d(5-t²)/dt = -2t.

Step 3: Use the chain rule to differentiate z with respect to t:

dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)

= (cos(x/y) · (1/y))(4) + (-sin(x/y) · (x/y²))(-2t)

= (4/y) · cos(4t/(5-t²)) + (2tx/y²) · sin(4t/(5-t²)).

This gives us the rate of change of z with respect to t.