Differentiation of vector-valued functions

Goal: Find the derivative of vector-valued functions by calculating the derivative of the real-valued functions of the vector-valued function.

Differentiation of vector-valued functions

Let's consider the vector-valued function r(t) = f(t)i + g(t)j or r(t) = f(t)i + g(t)j + h(t)k, we can find the derivative of these vector-valued  functions by calculating the derivative of the real-valued functions . The following theorem allows us to do that.

Theorem

Let f, g, h be differentiable functions of t:

Example

Use differentiation to find the derivative of the following vector-valued functions:

Solution

Let's use the formula of the differentiation of vector-valued functions and the formulas of the differentiation of real-valued functions

Practice

Calculate the derivative of the following function: