Recall that for a vector 𝑣 ∈ R^𝑚, the transposed vector is denoted 𝑣^T. Use the upper/lower notation for components respectively partial derivations (𝜕𝑓^𝑖)/(𝜕𝑦^𝑘)=: 𝑓^𝑖_𝑘 and (𝜕^2𝑓^𝑖)/(𝜕𝑦^𝑘𝜕𝑦^𝑙)=: 𝑓^𝑖_(𝑘𝑙) . We note by 𝑓_𝑦 the so-called Jacobian matrix of 𝑓 = 𝑓 (𝑦), that is the matrix with entries (𝑓_𝑦 )(𝑖𝑗) = 𝑓^𝑖_𝑗, then 𝑓^2_𝑦 is the matrix product and 𝑓(𝑦𝑦) has entries (𝑓^𝑖_(𝑘𝑙)). Consider a function 𝑓 : R^𝑚 → R^𝑚 𝑓 (𝑦) = (𝑓^1(𝑦^1, . . . , 𝑦^𝑚), . . . , 𝑓^𝑚 (𝑦^1, . . . , 𝑦^𝑚))^T. Show that the following identity holds (𝑓_𝑦 𝑓 )𝑦 𝑓 = 𝑓^T𝑓(𝑦𝑦) 𝑓 + 𝑓^2_𝑦𝑓