Derivative of Vector-Valued Function

The derivative of a scalar-valued function with respect to multiple inputs is the gradient.
The derivative of a vector-valued function with respect to an input is a tangent vector to the curve.

Let’s say a vector-valued function 𝑓 is defined as:

The derivative of 𝑓 over 𝑡 is defined as:

Partial Derivative of Vector-Valued Function

Let’s say a vector-valued function 𝑓 of vector input 𝐱 = (𝑥₁, 𝑥₂) is defined as:


The partial derivative of scalar value function 𝑓₁ over 𝐱 is defined as: $\nabla_{x} f_{1} = \frac{𝛿 f _{1}}{𝛿 x} = [\frac{𝛿 f _{1}}{𝛿 x _{1}} \frac{𝛿 f _{1}}{𝛿 x _{2}}]$	The partial derivative of 𝑓 over 𝑥₁ is defined as: $\frac{𝛿 f}{𝛿 x _{1}} = [\frac{𝛿 f _{1}}{𝛿 x _{1}} \frac{𝛿 f _{2}}{𝛿 x _{1}}]$
The partial derivative of scalar value function 𝑓₂ over 𝐱 is defined as: $\nabla_{x} f_{2} = \frac{𝛿 f _{2}}{𝛿 x} = [\frac{𝛿 f _{2}}{𝛿 x _{1}} \frac{𝛿 f _{2}}{𝛿 x _{2}}]$	The partial derivative of 𝑓 over 𝑥₂ is defined as: $\frac{𝛿 f}{𝛿 x _{2}} = [\frac{𝛿 f _{1}}{𝛿 x _{2}} \frac{𝛿 f _{2}}{𝛿 x _{2}}]$

$\nabla_{x} f_{1} = \frac{𝛿 f _{1}}{𝛿 x} = [\frac{𝛿 f _{1}}{𝛿 x _{1}} \frac{𝛿 f _{1}}{𝛿 x _{2}}]$

The partial derivative of 𝑓 over 𝑥₁ is defined as:

$\nabla_{x} f_{2} = \frac{𝛿 f _{2}}{𝛿 x} = [\frac{𝛿 f _{2}}{𝛿 x _{1}} \frac{𝛿 f _{2}}{𝛿 x _{2}}]$

The partial derivative of 𝑓 over 𝑥₂ is defined as:

This is used to build the Jacobian Matrix, and in this case is:

$J_{f} = \nabla_{x} f = [\frac{𝛿 f _{1}}{𝛿 x} \frac{𝛿 f _{2}}{𝛿 x}] = [\frac{𝛿 f _{1}}{𝛿 x _{1}} \frac{𝛿 f _{2}}{𝛿 x _{1}} \frac{𝛿 f _{1}}{𝛿 x _{2}} \frac{𝛿 f _{2}}{𝛿 x _{2}}]$