Final answer:
To find the partial derivatives ∆z/∆s and ∆z/∆t for the function z = ln(5x4y), with x and y given in terms of s and t, one must use the chain rule to combine the derivatives of z with respect to x and y with the derivatives of x and y with respect to s and t, respectively.
Step-by-step explanation:
The given function is z = ln(5x4y), where x = s*sin(t) and y = t*cos(s). To find ∆z/∆s and ∆z/∆t using the chain rule, we first differentiate z with respect to x and y, then we multiply each by the derivative of x and y with respect to s and t, respectively.
We have ∆z/∆x = 1/(5x4y) • 5•4x3 and ∆z/∆y = 1/(5x4y), by differentiating z with respect to x and y.
Next, differentiate x with respect to s and t: ∆x/∆s = sin(t) and ∆x/∆t = s*cos(t).
Similarly for y: ∆y/∆s = -t*sin(s) and ∆y/∆t = cos(s).
Now, apply the chain rule: ∆z/∆s = (∆z/∆x)(∆x/∆s) + (∆z/∆y)(∆y/∆s) and ∆z/∆t = (∆z/∆x)(∆x/∆t) + (∆z/∆y)(∆y/∆t).
By substituting and simplifying, we find the partial derivatives ∆z/∆s and ∆z/∆t.