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I have the following functions:

$f(x,y)=xy,\; g(t)=(e^t,\cos(t))$

And I want to calculate the derivative matrix of $f$ after $g.$

I calculate the jacobian matrix of $f$, which was a matrix, like

$$ D_f=\begin{bmatrix} y & x \end{bmatrix} $$

then on the components of $g$

$$ D_f(e^t,\cos(t))=\begin{bmatrix} \cos(t) & e^t \end{bmatrix} $$

then calculate the matrix of $g$

$$ D_g(e^t,\cos(t))=\begin{bmatrix} e^t \\ -\sin(t) \end{bmatrix} $$

then to calculate the matrix of the derivatives of $f$ after $g$, I multiplied the matrices

$$ D_f(e^t,\cos(t))=\begin{bmatrix} \cos(t) & e^t \end{bmatrix} $$

times

$$ D_g(e^t,\cos(t))=\begin{bmatrix} e^t \\ -\sin(t) \end{bmatrix}$$

$= \cos(t)e^t-e^t\sin(t).$

I have no way of checking if this is correct. Is this correct?

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Yes, this is correct. You can easily check this yourself. The composition is given by \begin{align} f(g(t))=f(e^{t},\cos (t))=e^{t}\cos(t). \end{align} If we differentiate this function we get \begin{align} (f(g(t))'=\frac{d}{dt}(e^t\cos(t))=e^t\cos (t)-e^t\sin(t) \end{align}

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