Properties of the derivatives of vector-valued functions

 Goal: Calculate the derivatives of vector-valued functions using the properties of derivatives

Properties of the derivatives of vector-valued functions

The constant multiple rule, the sum and difference rule, the product rule and the chain rule used in real valued functions are also applicable for vector-valued functions.

Theorem

Let r and u be differentiable vector-valued functions of t. Let f be a differentiable real-valued function of t and c a scalar.

In property IV, we use dot product and in property V cross product.

Example:

Calculate each of the derivatives using the properties of the derivatives of vector-valued functions.

Solution

a.

According to property IV (dot product), we have:

b. To solve this problem, we need to adapt it to property V (cross product). Property V is stated as follow:

Recall that the cross product of any vector with itself is zero. Let's calculate the second derivative of u(t):

Finally we have:

Practice