This function computes a coefficient in the (multivariate) Taylor series representation of expr without forming the series (it uses differentiation and substitution).  Often, expr is a polynomial.

The one-variable and several-variable cases are distinguished by the types of the input parameters.

UNIVARIATE CASE: x is a name and k a non-negative integer.

In this case, the value returned is the coefficient of ${\left(x-\mathrm{\alpha }\right)}^k$ in the Taylor series expansion of expr about $x=\mathrm{\alpha }$.  This is equivalent to executing  $\mathrm{coeff}\left(\mathrm{taylor}\left(\mathrm{expr}\,x=\mathrm{\alpha }\,k+1\right)\,x-\mathrm{\alpha }\,k\right)$  but it is more efficient (because only a single term is computed).

MULTIVARIATE CASE: x is a nonempty list $\left[x_1\,\dots \,x_v\right]$ of indeterminates appearing in expr and $\mathrm{\alpha }$ is a list $\left[{\mathrm{\alpha }}_1\,\dots \,{\mathrm{\alpha }}_v\right]$ specifying the point of expansion with respect to the given indeterminates;  k is a list $\left[k_1\,\dots \,k_v\right]$ of non-negative integers corresponding to elements in x and $\mathrm{\alpha }$.

In this case, the value returned is the coefficient of the term specified by the monomial

in the multivariate Taylor series expansion of expr about the point $x=\mathrm{\alpha }$.  If k is the list of zeros then the value returned is the value resulting from substituting $x=\mathrm{\alpha }$ into expr.

$p≔2x^2+3y^3-5$

$p≔3y^3+2x^2-5$

$\mathrm{coeftayl}\left(p\,x=0\,2\right)$

$2$

$\mathrm{coeftayl}\left(p\,x=1\,1\right)$

$4$

$\mathrm{taylor}\left(p\,x=1\right)$

$3y^3-3+4\left(x-1\right)+2{\left(x-1\right)}^2$

$q≔3a{\left(x+1\right)}^2+\sin \left(a\right)x^{2}y-y^{2}x+x-a$

$q≔3a{\left(x+1\right)}^2+\sin \left(a\right)x^{2}y-y^{2}x+x-a$

$\mathrm{coeftayl}\left(q\,x=-1\,2\right)$

$\sin \left(a\right)y+3a$

$\mathrm{coeftayl}\left(q\,x=-1\,1\right)$

$-2\sin \left(a\right)y-y^2+1$

$\mathrm{taylor}\left(q\,x=-1\right)$

$\sin \left(a\right)y+y^2-1-a+\left(-2\sin \left(a\right)y-y^2+1\right)\left(x+1\right)+\left(\sin \left(a\right)y+3a\right){\left(x+1\right)}^2$

$\mathrm{coeftayl}\left(q\,\left[x\,y\right]=\left[0\,0\right]\,\left[0\,0\right]\right)$

$2a$

$\mathrm{coeftayl}\left(q\,\left[x\,y\right]=\left[0\,0\right]\,\left[2\,1\right]\right)$

$\sin \left(a\right)$

$\mathrm{mtaylor}\left(q\,\left[x\,y\right]\right)$

$2a+\left(6a+1\right)x+3a{x}^2+\sin \left(a\right)x^{2}y-y^{2}x$

$\mathrm{coeftayl}\left(q\,\left[x\,y\right]=\left[0\,1\right]\,\left[1\,1\right]\right)$

$\mathrm{-2}$

$\mathrm{coeftayl}\left(q\,\left[x\,y\right]=\left[0\,1\right]\,\left[2\,1\right]\right)$

$\sin \left(a\right)$

$\mathrm{mtaylor}\left(q\,\left[x=0\,y=1\right]\right)$

$2a+6ax+\left(\sin \left(a\right)+3a\right)x^2-2\left(y-1\right)x+\sin \left(a\right)x^2\left(y-1\right)-{\left(y-1\right)}^{2}x$