Question

the partial derivative 4ye^(4xy)
the answer is 16xye^(4xy)+e^(4xy)(4). how do they get that answer please show work

Answer 1

implicit differentiation is my guess

Answer 2

f(x,y)=e^(4xy)
fx=4ye^(4xy) and fy=4xe^(4xy)
i dont understand how they got fxy?

Answer 3

\[\begin{align*}\frac{\partial}{\partial y}4ye^{4xy}&=4\frac{\partial}{\partial y}[y]e^{4xy}+4y\frac{\partial}{\partial y}[e^{4xy}]\\
&=4e^{4xy}+4ye^{4xy}\cdot\frac{\partial}{\partial y}[4xy]\\
&=4e^{4xy}+4ye^{4xy}(4x)\\
&=4e^{4xy}+16xye^{4xy}\end{align*}\]
In order, that would be product rule and chain rule, then simplification.

Answer 4

thank you