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### Formal implicit differentiation

I am not perfectly sure that the following code evaluate the implicit differentiation of the function UU. In all case what I expect is to fint $\partial w_2/ \partial w_1$ and then $\partial w_2^2/ \partial^2 w_1$

U=function('U')(x)
w2=function('w2')(w1)
U=U(x)
UU=p*U(w1)+(1-p)*U(w2)
diff(UU, w1)


then I would like to substitute $U(x) = log(x)$ or $U(x) = x^a$ or $U(x) = -\exp(-ax)$. And as an implied question, I would like to know how to select part of an expression like for instance $\exp(ax + b)+ f(x) -> a, \text{ or } a*x \text{ or } f(x)$. As every expression in Mathematica is a three, is not difficult to saw a branch. But in Sagemath I wonder how to do.

### Formal implicit differentiation

I am not perfectly sure that the following code evaluate the implicit differentiation of the function UU. In all case what I expect is to fint $\partial w_2/ \partial w_1$ and then $\partial w_2^2/ \partial^2 w_1$

U=function('U')(x)
w2=function('w2')(w1)
U=U(x)
UU=p*U(w1)+(1-p)*U(w2)
diff(UU, w1)


then I would like to substitute $U(x) = log(x)$ or $U(x) = x^a$ or $U(x) = -\exp(-ax)$. And as an implied question, I would like to know how to select part of an expression like for instance $\exp(ax + b)+ f(x) -> a, \text{ or } a*x \text{ or } f(x)$. As every expression in Mathematica is a three, is not difficult to saw a branch. But in Sagemath I wonder how to do.

### Formal implicit differentiation

I am not perfectly sure that the following code evaluate the implicit differentiation of the function UU. In all case what I expect is to fint $\partial w_2/ \partial w_1$ and then $\partial w_2^2/ \partial^2 w_1$

U=function('U')(x)
w2=function('w2')(w1)
U=U(x)
UU=p*U(w1)+(1-p)*U(w2)
diff(UU, w1)


### Formal implicit differentiation

I am not perfectly sure that the following code evaluate the implicit differentiation of the function UU. In all case what I expect is to fint $\partial w_2/ \partial w_1$ and then $\partial w_2^2/ \partial^2 w_1$

U=function('U')(x)
w2=function('w2')(w1)
U=U(x)
UU=p*U(w1)+(1-p)*U(w2)
diff(UU, w1)


then I would like to substitute $U(x) = \log(x)$ or $U(x) = x^a$ or $U(x) = -\exp(-a x)$. And as an implied question, I would like to know how to select part of an expression like for instance $\exp(ax + b)+ f(x)$ -> , $a$, or $ax$ or $f(x)$. $f(x)$. As every expression in Mathematica is a three, is not difficult to saw a branch. But in Sagemath I wonder how to do.

### Formal implicit differentiation

I am not perfectly sure that the following code evaluate the implicit differentiation of the function UU. In all case what I expect is to fint find $\partial w_2/ \partial w_1$ and then $\partial w_2^2/ \partial^2 w_1$

U=function('U')(x)
w2=function('w2')(w1)
U=U(x)
UU=p*U(w1)+(1-p)*U(w2)
diff(UU, w1)


then I would like to substitute $U(x) = \log(x)$ or $U(x) = x^a$ or $U(x) = -\exp(-a x)$. And as an implied question, I would like to know how to select part of an expression like for instance $\exp(ax + b)+ f(x)$ , $a$, or $ax$ or $f(x)$. As every expression in Mathematica is a three, is not difficult to saw a branch. But in Sagemath I wonder how to do.

### Formal implicit differentiation

I am not perfectly sure that the following code evaluate the implicit differentiation of the function UU. In all case what I expect is to find $\partial w_2/ \partial w_1$ and then $\partial w_2^2/ \partial^2 w_1$$\partial^2 w_2/ \partial w_1^2$

U=function('U')(x)
w2=function('w2')(w1)
U=U(x)
UU=p*U(w1)+(1-p)*U(w2)
diff(UU, w1)


then I would like to substitute $U(x) = \log(x)$ or $U(x) = x^a$ or $U(x) = -\exp(-a x)$. And as an implied question, I would like to know how to select part of an expression like for instance $\exp(ax + b)+ f(x)$ , $a$, or $ax$ or $f(x)$. As every expression in Mathematica is a three, is not difficult to saw a branch. But in Sagemath I wonder how to do. 10 None

I am not perfectly sure that the following code evaluate evaluates the implicit differentiation of the function UU. In all any case what I expect is to find $\partial w_2/ \partial w_1$ and then $\partial^2 w_2/ \partial w_1^2$w_1^2$. U=function('U')(x) w2=function('w2')(w1) U=U(x) UU=p*U(w1)+(1-p)*U(w2) U = function('U')(x) w2 = function('w2')(w1) U = U(x) UU = p*U(w1) + (1 - p)*U(w2) diff(UU, w1)  then I would like to substitute$U(x) U(x) = \log(x)$log(x) or$U(x) U(x) = x^a$x^a or$U(x) U(x) = -\exp(-a x)$. -exp(-a*x). And as an implied question, I would like to know how to select part of an expression like for instance$\exp(ax instanceexp(a*x + b)+ f(x)$b) + f(x),$a$, a, or a*x or$ax$or$f(x)$. f(x). As every expression in Mathematica is a three, tree, is not difficult to saw a branch. But in Sagemath SageMath I wonder how to do. ### Formal implicit differentiation I am not perfectly sure that the following code evaluates the implicit differentiation of the function UU. In any case what I expect is to find$\partial w_2/ \partial w_1$and then$\partial^2 w_2/ \partial w_1^2$. U = function('U')(x) w2 = function('w2')(w1) U = U(x) UU = p*U(w1) + (1 - p)*U(w2) diff(UU, w1)  then I would like to substitute U(x) = log(x) or U(x) = x^a or U(x) = -exp(-a*x). And as an implied question, I would like to know how to select part of an expression like for instanceexp(a*x + b) + f(x), a, or a*x or f(x). As every expression in Mathematica is a tree, it is not difficult to saw a branch. But in SageMath I wonder how to do.do this type of operation. ### Formal implicit differentiation As my question was poorly asked, I am not perfectly sure that the following code evaluates the implicit differentiation of the function UU. In any case what I expect is to find$\partial w_2/ \partial w_1$and then$\partial^2 w_2/ \partial w_1^2\$.rewrite it.

U = function('U')(x)
w2 = function('w2')(w1)
U = U(x)
UU = p*U(w1) + (1 - p)*U(w2)
diff(UU, w1)


then I would like to substitute U(x) = log(x) or U(x) = x^a or U(x) = -exp(-a*x). And as an implied question, I would like to know how to select part was searching for the code of first order and second order differentiation of an expression like for instanceexp(a*x + b) + f(x), a, or a*ximplicit function. Finally I found or f(x). As every expression in Mathematica is a tree, it is not difficult to saw a branch. But in SageMath I wonder how to do this type of operation.and post it.