Introduction to the Derivative of Vector-Valued Function

Calculus is the study of continuous rate of change which allows us to find the rate of instantaneous functions like a vector-valued function. But the question is how we can find the derivative of a vector function since it has a different number of components like i, j and k. Let's understand how to find derivatives of a vector function.

Understanding of the Derivative of Vector-valued Functions

A vector-valued function is a function of one or more variables whose range is a set of multidimensional vectors. It is also known as vector function. This kind of function has magnitude as well as direction. In calculus, a vector-valued function is used to determine a curve in space as the collection of terminal points. If the curve is smooth, then it is possible that it has a derivative.

The definition of the derivative of a vector-valued function is similar to the definition of a real-valued function of one variable. But since the range of a vector-valued function consists of a set of vectors, it will be true for the range of derivatives of a vector-valued function.

Derivative of Vector-Valued Function

The first principle of derivative is used to define the derivative of a vector function. This method is also known as the delta method. For a function f(x), the principle of derivative is given by:

$f'(x)=\lim_{\Delta t\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$

Where f’(x) is the derivative of f(x). Now we will define the same definition for a vector function derivative. For this, suppose that r(t) is a differentiable vector function defined on an open interval (a, b), the derivative of r(t) will be expressed as:

$\vec{r}(t)=\lim_{\Delta t\to 0}\frac{\vec{r}(t+\Delta t)-\vec{r}(t)}{t}$

If r(t) is defined on a closed interval [a, b], then the following limits must exists as well:

$\vec{r}'(a)=\lim_{\Delta t\to 0+}\frac{\vec{r}(a+\Delta t)-\vec{r}(a)}{\Delta t}$

And,

$\vec{r}'(b)=\lim_{\Delta t\to 0-}\frac{\vec{r}(b+\Delta t)-\vec{r}(b)}{\Delta t}$

Since the same principle is used to define derivatives of a vector function, we can assume it as an instantaneous rate of change as well. Therefore, all derivative rules can be also defined to calculate derivatives of a vector-valued function. Let us understand the rules to calculate derivatives of a vector function. 

Rules of the derivative of Vector-valued functions

There are six rules of derivatives for a vector-valued function. For two vector-valued function r and u, we can define all derivative rules such as:

1. If c is a scalar multiplied with r(t) then the scalar multiple rule is,
   $\frac{d}{dt}[c\vec{r}(t)]=c\vec{r}'(t)$
2. If r(t) and u(t) are added or subtracted together, the sum and difference rule of derivative is,
   $\frac{d}{dt}[\vec{r}(t)\pm\vec{u}(t)]=\vec{r}'(t)\pm\vec{u}'(t) $
3. If there is a product of r(t) and u(t), the product rule of derivative is,
   $\frac{d}{dt}[\vec{r}(t)\vec{u}(t)]=\vec{r}'(t)\vec{u}(t)+\vec{r}(t)\vec{u}'(t)$
4. If there is a scalar function multiplied with a vector function, the scalar product rule is,
   $\frac{d}{dt}[f(t).\vec{r}(t)]=f'(t)\vec{r}(t)+f(t)\vec{r}'(t)$
5. If a vector function is combined with another function, the chain rule of derivative is,
   $\frac{d}{dt}[\vec{r}(f(t))]=\vec{r}'(f(t)).f'(t)$
6. If a vector function is multiplied with itself, it results in a constant function. Then the, derivative of the vector function will be,
   $\vec{r}(t).\vec{r}(t)=c\quad then,\quad \vec{r}(t).\vec{r}(t)=0$

How to calculate the derivative of a vector-valued function?

To calculate the derivative of a vector function, we need to follow the given steps.

1. Identify the vector-valued function.
2. Apply the derivative first principle and use derivative rules according to the function involved.
3. The derivative will be applied on everly component of the vector function such as i, j and k. 
4. Simplify if needed. 

Let us understand how to compute derivatives of a vector function in the following examples.

Derivative of a Vector-valued Function example

Compute the derivative of the given function by using first principle.

$\vec{r}(t)=(3t+4)i+(t^2-4t+3)j$

There are two methods to calculate derivatives of this function. First one is to use the first principle method and second is to use ordinary derivative rules. So, we will use first principle which is,

Hence the derivative of the given vector function is 3i + (2t - 4)j. 

Conclusion

The derivative of a vector-valued function is the rate of change of a vector function whose domain is a set of vectors. The calculation of the derivative of a vector is similar to the first principle of derivative. Therefore, the first principle is used to compute derivatives. Moreover, it also follows all of the derivative rules such as power rule, chain rule, product rule etc.