Answer :
The expression representing the partial derivative ∂²P/∂z∂y is given by ∂²P/∂z∂y = cosh(|z|) multiplied by a constant factor of 0, which simplifies to 0.
To find the expression representing the partial derivative ∂²P/∂z∂y, we differentiate P(z, y) = 2 + sinh(|z|) + ln(25) with respect to z and y separately.
Taking the derivative with respect to z, we consider that sinh(|z|) is an odd function and its derivative will have the same property. Therefore, the derivative of sinh(|z|) with respect to z will be cosh(|z|) multiplied by the derivative of |z| with respect to z, which is either 1 or -1 depending on the sign of z.
Since we have absolute value signs around z, we need to consider both cases. Hence, the partial derivative of sinh(|z|) with respect to z will be cosh(|z|) if z > 0 and -cosh(|z|) if z < 0.
Next, taking the derivative with respect to y, the term ln(25) is a constant and its derivative will be zero. Therefore, the partial derivative of ln(25) with respect to y is zero.
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