find all second order derivatives for ( z = 7ye^{6x} ).\n( 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})