Question

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

Explanation:

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

Using the chain rule \((e^{u})^\prime=e^{u}\cdot u^\prime\) (where \(u = 6x\)), and treating \(y\) as a constant.
\(z_{x}=\frac{\partial z}{\partial x}=7y\cdot\frac{\partial(e^{6x})}{\partial x}\)
Since \(\frac{\partial(e^{6x})}{\partial x}=6e^{6x}\), then \(z_{x}=42ye^{6x}\)

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

Differentiate \(z_{x}=42ye^{6x}\) with respect to \(x\) again.
Using the chain rule \((e^{u})^\prime=e^{u}\cdot u^\prime\) (where \(u = 6x\)) and treating \(y\) as a constant.
\(z_{xx}=\frac{\partial z_{x}}{\partial x}=42y\cdot\frac{\partial(e^{6x})}{\partial x}\)
Since \(\frac{\partial(e^{6x})}{\partial x}=6e^{6x}\), then \(z_{xx}=252ye^{6x}\)

Answer:

\(252ye^{6x}\)