# How to put derivative of composition in Jacobian matrix?

Here are two functions: $f\left(u,v\right)=u^{2}+3v^{2}$

$g\left(x,y\right)=\begin{pmatrix} e^{x}\cos y \\ e^{x}\sin y \end{pmatrix}$

I need to make Jacobian matrix of $f\circ g$. I found derivative of their composition:

$\frac{d\left(f\circ g\right) }{d\left(x,y\right) }=2e^{2x}\cos^{2}{y}+4e^{2x}\sin{y}\cos{y}+6e^{2x}sin^{2}{y}$

How do I put that in Jacobian matrix?

• What is $\frac{d(f\circ g)}{d(x,y)}$? Try to understand your lesson before getting stuck in exercises. Jun 11 '16 at 13:11
• If I understood it I wouldnt post here. Jun 11 '16 at 13:15
• In this case you should post a question about what you didn't understand in the lesson ;) Jun 11 '16 at 13:16

• The last line is indeed a $1\times 2$ matrix. Just as in @Bye-World's answer Jun 11 '16 at 13:26
• I just played around with the trigonometric identity $\sin^2 y+\cos^2 y=1$ Jun 11 '16 at 13:30
$$(f\circ g)(x,y) = h(x,y) = e^{2x}\cos^2(y)+3e^{2x}\sin^2(y)$$ Now just build the Jacobian matrix (AKA gradient because $h$ is a scalar-valued function) like normal: $$\pmatrix{\frac{\partial h}{\partial x} & \frac{\partial h}{\partial y}}$$