15  Derivatives

TODO

Need to mention that the following notes are always assuming the function is continuous at all points.

Functions

Scalar-valued and vector-valued functions

A scalar-valued function f outputs a scalar, while a vector-valued function \mathbf{f} outputs a vector.

A vector-valued function can be interpreted as a vector of scalar-valued functions

\mathbf{f} = \begin{bmatrix} f_{1} \\ \vdots \\ f_{m} \\ \end{bmatrix}.

Univariate and multivariate functions

A univariate function takes a scalar x as the input, while a multivariate function takes a vector \mathbf{x} as the input

Scalar-valued univariate (ordinary) derivatives

The derivative function of a univariate scalar-valued function f, denoted \frac{ d f }{ d x } or f', is another univariate scalar-valued function that takes x as the input and outputs the derivative of the function at point x:

f' (x) = \frac{ d f }{ d x } (x) = \lim_{h \rightarrow 0} \frac{ f (x + h) - f (x) }{ h }.

The derivative of f at point x is the slope of the tangent line to the function f at the point x, which also represents the rate of the change of f at x in the positive (increasing) direction.

A line has the form of

y = k x + b

where k is the slope of the line and b represents the interception of the line on the y axis.

We can calculate the slope of the line if we know the two points (x_{1}, y_{1}) and (x_{2}, y_{2}) that are on the line

\begin{aligned} y_{2} - y_{1} & = k x_{2} + b - k x_{1} + b \\ & = k (x_{2} - x_{1}) \\ k & = \frac{ y_{2} - y_{1} }{ x_{2} - x_{1} }. \end{aligned}

Thus, if we set x' = x + h, the equation

\frac{ f (x + h) - f (x) }{ h } = \frac{ f (x') - f(x) }{ x' - x }

represents the the slope of the line connecting the points (x', f (x')) and (x, f(x)).

As h getting smaller, the line connecting the points (x', f (x')) and (x, f(x)) is becoming the tangent line of f(x) at point x.

Scalar-valued multivariate derivatives

Directional derivatives

Given a non-zero vector \mathbf{u} \in \mathbb{R}^{n}, the directional derivative function of a scalar-valued multivariate function f, denoted D_{\mathbf{u}} f, is another scalar-valued multivariate function that takes \mathbf{x} \in \mathbb{R}^{n} as the input and outputs the directional derivative of the function with respect to \mathbf{u} at point \mathbf{x}:

D_{\mathbf{u}} f (\mathbf{x}) = \lim_{h \rightarrow 0} \frac{ f (\mathbf{x} + h \mathbf{u}) - f (\mathbf{x}) }{ h \lVert \mathbf{u} \rVert }.

If the given vector \mathbf{u} is a unit vector (\lVert \mathbf{u} \rVert = 1), the directional derivative function is simplified to

D_{\mathbf{u}} f (\mathbf{x}) = \lim_{h \rightarrow 0} \frac{ f (\mathbf{x} + h \mathbf{u}) - f (\mathbf{x}) }{ h }.

The directional derivative of f with respect to the vector \mathbf{u} at the point \mathbf{x} represents the rate of change of f at \mathbf{x} in the positive direction of \mathbf{u}.

Reparameterization of directional derivatives

Given the vectors \mathbf{x} and \mathbf{u}, the univariate function

g (\epsilon) = f (\mathbf{x} + \epsilon \mathbf{u})

describes the values of f along a particular line that starts from the point \mathbf{x} and goes in the direction of the vector \mathbf{u}.

The derivative of function g at point \epsilon = 0 gives the rate of change of g at point 0 and also the rate of change of f at \mathbf{x} in the direction of \mathbf{u}, which is exactly the directional derivative:

D_{u} f (\mathbf{x}) \lVert \mathbf{u} \rVert = \frac{ d }{ d \epsilon } g (\epsilon) \Big|_{\epsilon = 0} = \frac{ d }{ d \epsilon } f (\mathbf{x} + \epsilon \mathbf{u}) \Big|_{\epsilon = 0}.

\begin{aligned} \frac{ d g }{ d \epsilon } (\epsilon) \Big|_{\epsilon = 0} & = \lim_{h \rightarrow 0} \frac{ c (\epsilon + h) - c (\epsilon) }{ h } \Big|_{\epsilon = 0} \\ & = \lim_{h \rightarrow 0} \frac{ f (\mathbf{x} + (\epsilon + h) \mathbf{u}) - f (\mathbf{x} + \epsilon \mathbf{u}) }{ h } \Big|_{\epsilon = 0} \\ & = \lim_{h \rightarrow 0} \frac{ f (\mathbf{x} + h \mathbf{u}) - f (\mathbf{x}) }{ h } \\ & = D_{\mathbf{u}} f (\mathbf{x}) \lVert \mathbf{u} \rVert. \end{aligned}

This observation provides an alternative way to compute a directional derivative that only requires computing an ordinary derivative.

Partial derivatives

The partial derivative function of a multivariate function f with respect to a dimension i, denoted as \frac{ \partial f }{ \partial x_{i} }, is a special case of the directional derivative function, where the given vector \mathbf{u} must be a standard basis vector \mathbf{e}_{i} of the dimension i:

\frac{ \partial f }{ \partial x_{i} } (\mathbf{x}) = D_{\mathbf{e}_{i}} f (\mathbf{x}) = \lim_{h \rightarrow 0} \frac{ f (\mathbf{x} + h \mathbf{e}_{i}) - f (\mathbf{x}) }{ h }.

The partial derivative of f with respect to \mathbf{e}_{i} at \mathbf{x} gives the rate of the change of the function at \mathbf{x} along the direction of the ith dimension of the vector space.

The partial derivative \frac{ \partial f }{ \partial x_{i} } (\mathbf{x}) can be calculated using the ordinary derivative

\frac{ \partial f }{ \partial x_{i} } (\mathbf{x}) = \frac{ d f |_{\mathbf{x}} }{ d x_{i} } (x_{i})

where f |_{\mathbf{x}} (x_{i}) is a univariate function obtained by setting all variables except x_{i} in f as constants.

Gradients

The gradient function of a multivariate function f, denoted as \nabla f, is a vector-valued function, where each element in the output vector is the partial derivative of f with respect to all standard basis vectors

\nabla f (\mathbf{x}) = \begin{bmatrix} \frac{ \partial f }{ \partial x_{1} } (\mathbf{x}) \\ \vdots \\ \frac{ \partial f }{ \partial x_{n} } (\mathbf{x}) \end{bmatrix}.

The directional derivative with respect to any non-zero vector \mathbf{u} of function f at point \mathbf{x} can be readily calculated using its gradient as follows:

D_{\mathbf{u}} f (\mathbf{x}) = \nabla f (\mathbf{x}) \cdot \frac{ \mathbf{u} }{ \lVert \mathbf{u} \rVert }.

Using the Reparameterization of directional derivative, we can write the directional derivative as

D_{\mathbf{u}} f (\mathbf{x}) \lVert \mathbf{u} \rVert = \frac{ d f }{ d \epsilon } (\mathbf{x} + \epsilon \mathbf{u}) \Big|_{\epsilon = 0}.

By using the chain rule and writing \mathbf{g} (\epsilon) = \mathbf{x} + \epsilon \mathbf{u}, we can also see that the derivative of the univariate function f (\mathbf{x} + \epsilon \mathbf{u}) at point \epsilon = 0 can be written as

\begin{aligned} \frac{ d f }{ d \epsilon } (\mathbf{x} + \epsilon \mathbf{u}) \Big|_{\epsilon = 0} & = \sum_{i=1}^{m} \frac{ \partial f }{ \partial g_{i} (\epsilon) } (\mathbf{g} (\epsilon)) \frac{ d g_{i} }{ d \epsilon } (\epsilon) \Big|_{\epsilon = 0} \\ & = \sum_{i=1}^{m} \frac{ \partial f }{ \partial (x_{i} + \epsilon u_{i}) } (\mathbf{x} + \epsilon \mathbf{u}) \frac{ d (x_{i} + \epsilon u_{i}) }{ d \epsilon } (\epsilon) \Big|_{\epsilon = 0} \\ & = \sum_{i=1}^{m} \frac{ \partial f }{ \partial (x_{i} + \epsilon u_{i}) } (\mathbf{x} + \epsilon \mathbf{u}) u_{i} \Big|_{\epsilon = 0} \\ & = \sum_{i=1}^{m} \frac{ \partial f }{ \partial x_{i} } (\mathbf{x}) u_{i} \\ & = \nabla f (\mathbf{x}) \cdot \mathbf{u}. \end{aligned}

Thus,

\begin{aligned} D_{\mathbf{u}} f (\mathbf{x}) = \nabla f (\mathbf{x}) \cdot \frac{ \mathbf{u} }{ \lVert \mathbf{u} \rVert }. \end{aligned}

It turns out that the directional derivative of function f at point \mathbf{x} with respect to the direction of the gradient vector is the largest. That is, the gradient of f at point \mathbf{x} is the direction of steepest ascent along the function surface at the point \mathbf{x}

\nabla f (\mathbf{x}) = \arg\max_{\mathbf{u}} D_{\mathbf{u}} f (\mathbf{x}).

According to the definition of a dot product,

\begin{aligned} D_{\mathbf{u}} f (\mathbf{x}) & = \frac{ \nabla f (\mathbf{x}) \cdot \mathbf{u} }{ \lVert \mathbf{u} \rVert } \\ & = \frac{ \lVert \nabla f (\mathbf{x}) \rVert \lVert \mathbf{u} \rVert \cos (\theta) }{ \lVert \mathbf{u} \rVert } \\ & = \lVert \nabla f (\mathbf{x}) \rVert \cos (\theta) \end{aligned}

where \theta is the angle between the vector \nabla f(x) and \mathbf{u}.

Since \lVert \nabla f (\mathbf{x}) \rVert is fixed given \mathbf{x},

\max_{\mathbf{u}} D_{\mathbf{u}} f (\mathbf{x}) = \max_{\theta} \cos (\theta),

which is maximized when \mathbf{u} has the same direction as \nabla f (\mathbf{x}).

The directional derivative of the function f with respect to the direction of the gradient vector \nabla f (\mathbf{x}) at point \mathbf{x} is the magnitude of the gradient vector

D_{\nabla f (\mathbf{x})} f (\mathbf{x}) = \lVert \nabla f (\mathbf{x}) \rVert.

Replacing the \mathbf{u} with \nabla f (\mathbf{x}) in the equation

D_{\mathbf{u}} f (\mathbf{x}) = \nabla f (\mathbf{x}) \cdot \frac{ \mathbf{u} }{ \lVert \mathbf{u} \rVert }

to get

\begin{aligned} D_{\nabla f (\mathbf{x})} f (\mathbf{x}) & = \nabla f (\mathbf{x}) \cdot \frac{ \nabla f (\mathbf{x}) }{ \lVert \nabla f (\mathbf{x}) \rVert } \\ & = \frac{ \lVert \nabla f (\mathbf{x}) \rVert^{2} }{ \lVert \nabla f (\mathbf{x}) \rVert } \\ & = \lVert \nabla f (\mathbf{x}) \rVert. \end{aligned}

Derivatives for vector-valued functions

Since the vector-valued functions are just the vector version of their corresponding scalar-valued functions, all types of derivatives for vector-valued functions are the vectorized versions of their corresponding derivatives for the scalar-valued functions.

  • Ordinary derivative functions

    \mathbf{f}' (x) = \frac{ \mathop{d} \mathbf{f} }{ \mathop{d} x } (x) = \begin{bmatrix} f_{1}' (x) \\ \vdots \\ f_{m}' (x) \\ \end{bmatrix}.

  • Directional derivative functions

    D_{\mathbf{u}} \mathbf{f} (\mathbf{x}) = \begin{bmatrix} D_{\mathbf{u}} f_{1} (\mathbf{x}) \\ \vdots \\ D_{\mathbf{u}} f_{m} (\mathbf{x}) \\ \end{bmatrix}.

  • Jacobian matrix, whose row i is the transposed gradient of the scalar-valued function f_{i} w.r.t \mathbf{x}

    \mathbf{J} \mathbf{f} (\mathbf{x}) = \begin{bmatrix} - & \nabla^{T} f_{1} (\mathbf{x}) & - \\ & \vdots & \\ - & \nabla^{T} f_{m} (\mathbf{x}) & - \\ \end{bmatrix} = \begin{bmatrix} \frac{ \partial f_{1} }{ \partial x_{1} } (\mathbf{x}) & \dots & \frac{ \partial f_{1} }{ \partial x_{n} } (\mathbf{x}) \\ \vdots & \ddots & \vdots \\ \frac{ \partial f_{m} }{ \partial x_{1} } (\mathbf{x}) & \dots & \frac{ \partial f_{m} }{ \partial x_{n} } (\mathbf{x}) \\ \end{bmatrix} = \begin{bmatrix} | & & | \\ \nabla f_{1} (\mathbf{x}) & \dots & \nabla f_{m} (\mathbf{x}) \\ | & & | \\ \end{bmatrix}^{T}.

Functional Derivatives

Function of functions

A function f itself can be viewed as an infinite-dimensional vector

D_{u} F [f] = \lim_{h \rightarrow 0} \frac{ F [f + h u] - F [f] }{ h }

Using a similar trick used in equation , we can see that

\begin{aligned} D_{u} F [f] & = D_{u} F [f + \epsilon u] \Big|_{\epsilon = 0} \\ & = \lim_{h \rightarrow 0} \frac{ F [f + \epsilon u + h u] - F [f + \epsilon u] }{ h } \Big|_{\epsilon = 0} \\ & = \lim_{h \rightarrow 0} \frac{ F [f + (\epsilon + h) u] - F [f + \epsilon u] }{ h } \Big|_{\epsilon = 0} \\ & = \frac{ d }{ d \epsilon } F [f + \epsilon u] \Big|_{\epsilon = 0} \\ \end{aligned}