Compute the derivatives df/dx of the following functions. Describe your steps in detail.

a. Use the chain rule, Provide the dimensions of every single partial derivative.

f(z) = exp(-1/2z)

z = g(y) = y^{T}s^{–}¹y

y=h(x) = x-μ

where x, μ∈R^{D}, S∈R^{DxD}.

b. Use the chain rule, Provide the dimensions of every single partial derivative. You do not need to compute the product of the partial derivatives explicitly.

f = tanh(z)∈R^{M}

z = Ax+b, x∈R^{N}, A∈R^{MxN}, b∈R^{M}

Here, tanh is applied to every component of z.

Answer:

A.

B.