Question

find all second order derivatives for ( z = 7ye^{6x} ).
( z_{xx}=252ye^{6x} )
( z_{yy}=)

Explanation:

Step1: Find the first - order derivative with respect to \(y\)

Given \(z = 7ye^{6x}\).
Using the rule \(\frac{\partial}{\partial y}(ay)=a\) (where \(a = 7e^{6x}\)), we have \(z_y=\frac{\partial z}{\partial y}=7e^{6x}\).

Step2: Find the second - order derivative with respect to \(y\)

Differentiate \(z_y = 7e^{6x}\) with respect to \(y\).
Since \(\frac{\partial}{\partial y}(c)=0\) (where \(c = 7e^{6x}\) is a constant with respect to \(y\)), we get \(z_{yy}=\frac{\partial^2 z}{\partial y^2}=0\).

Answer:

\(z_{yy}=0\)