= U(x,y) || || |- | A is not a function of x, g(X) is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. eX, sin(X), cos(X), ln(X), etc.); g(x) is the equivalent scalar function, g′(x) is its derivative, and g′(X) is the corresponding matrix function || || colspan=2| |- | A is not a function of x || || |} Further see Derivative of the exponential map. Scalar-by-scalar identities With vectors involved {|class="wikitable" style="text-align: center;" |+ Identities: scalar-by-scalar, with vectors involved ! scope="col" width="150" | Condition ! scope="col" width="10" | Expression ! scope="col" width="150" | Any layout (assumes dot product ignores row vs. column layout) |- | u = u(x) || || |- | u = u(x), v = v(x) || | colspan=2| |} With matrices involved {|class="wikitable" style="text-align: center;" |+Identities: scalar-by-scalar, with matrices involved ! scope="col" width="175" | Condition ! scope="col" width="100" | Expression ! scope="col" width="100" | Consistent numerator layout,i.e. by Y and XT ! scope="col" width="100" | Mixed layout,i.e. by Y and X |- | U = U(x) || || colspan=2| |- | U = U(x) || || colspan=2| |- | U = U(x) || | colspan=2 | |- | U = U(x) || || || |- | A is not a function of x, g(X) is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. eX, sin(X), cos(X), ln(X), etc.); g(x) is the equivalent scalar function, g′(x) is its derivative, and g′(X) is the corresponding matrix function. || || colspan=2| |- | A is not a function of x || || colspan=2| |} Identities in differential form It is often easier to work in differential form and then convert back to normal derivatives. This only works well using the numerator layout. In these rules, "a" is a scalar. {|class="wikitable" style="text-align: center;" |+ Differential identities: scalar involving matrix ! Condition !! Expression !! Result (numerator layout) |- | || || |- | || || |- | || || |} {|class="wikitable" style="text-align: center;" |+ Differential identities: matrix ! Condition !! Expression !! Result (numerator layout) |- |A is not a function of X || || |- |a is not a function of X || || |- | || || |- | || || |- | (Kronecker product) || || |- | (Hadamard product) || || |- | || || |- | | |