How do you use the chain rule to find dz/ds and dz/dt for the function z = x*y^5*x*s^2*t*y*s*t^2?

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Understanding the Problem

To find the derivatives dz/ds and dz/dt for the function z = x*y^5*x*s^2*t*y*s*t^2, we first need to clarify the relationships between the variables involved. The function appears to have multiple variables, and we will employ the chain rule to differentiate it effectively.

Step 1: Applying the Chain Rule

The chain rule states that if a variable z depends on variables s and t, we can express the derivative of z with respect to s as:

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

Similarly, for t, the derivative is given by:

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

Step 2: Find the Partial Derivatives

Let’s break down the function further to identify how z varies with respect to s and t.

1. **Re-arranging the function**: The given function can be rewritten by isolating the components involving s and t:

z = x * y^5 * s^2 * t * y * s * t^2

2. **Calculating Partial Derivatives**:

  • dz/ds: Treating t as constant, differentiate z with respect to s:

    dz/ds = x * y^5 * (2*s*t*y*s*t^2) + x * y^5 * (s^2*t*y)'s
  • dz/dt: Again, treat s as constant, differentiate z with respect to t:

    dz/dt = x*y^5*s^2*y*s*(t^2) + x*y^5*s^2*y*t*(2)

Step 3: Conclusion

Using the chain rule effectively allows us to evaluate how changes in s and t influence the function z. The calculations simplify depending on the known values of x and y. Finally, substituting specific values for these variables will allow for numerical results, providing further insight into the function’s behavior.

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