How do you apply the chain rule to find dz/dt for the function z = sin(x)cos(y) where x and y are functions of t, specifically x = t and y = 5t?

To find dz/dt using the chain rule, we first need to understand how z depends on x and y, and how these, in turn, depend on t. Given the function:

z = sin(x)cos(y)

where x = t and y = 5t, we can define the partial derivatives of z with respect to x and y.

Step 1: Find dz/dx and dz/dy

Using basic calculus, we calculate:

dz/dx = cos(x)cos(y)
dz/dy = -sin(x)sin(y)

Step 2: Calculate dx/dt and dy/dt

Now, since x and y are both functions of t, we also need their derivatives:

dx/dt = 1
dy/dt = 5

Step 3: Apply the Chain Rule

According to the chain rule, we find:

dz/dt = (dz/dx)*(dx/dt) + (dz/dy)*(dy/dt)

Substituting the values we have:

dz/dt = (cos(x)cos(y))(1) + (-sin(x)sin(y))(5)

This simplifies to:

dz/dt = cos(x)cos(y) – 5sin(x)sin(y)

Step 4: Substitute x and y back

Since we have x = t and y = 5t, we can put these back into our final equation:

dz/dt = cos(t)cos(5t) – 5sin(t)sin(5t)

Conclusion

Thus, the derivative of z with respect to t, using the chain rule, is: