# What are linear functions?

Linear functions are those which have a power of one. Given y = f(x), f(x) is a linear function of x if it contains powers of x not greater than 1. The straight lines are the best examples of linear functions and taking the **derivatives of linear functions **is as easy as nuts. We will see how to find **derivative of linear function** in detail.

## Method to find the derivative of a linear function:

Consider the basic linear function equation for a line y = f(x) = mx + b, for m and b being arbitrary real numbers. m is also called the slope of that line. We know that tangent to any straight line is itself the line. So computing the **derivative of a linear function **gives us the slope of the tangent line. It implies that the derivative of a function having a graph of a line with slope m is the constant m itself.

Lets compute the derivation results for above function. We already have the formula for finding the derivative as Here f(x) = mx + b => f(x + h) = m(x + h) + b = mx + mh + b. Taking the difference, f(x + h) – f(x) = mx + mh + b – mx – b = mh. Now f^{'}'(x) = [ ((f(x + h) – f(x))/ h]_{h->0}. Hence f^{'}'(x) = [mh / h]_{h->0} = m which is a constant.

Some linear functions are also piecewise which means that they are not continuous and have different slope at different instances. The **derivative of piecewise linear function **can be found by calculating the pieces of anti-derivatives and putting the initial value such that the integration constant gets eliminated.

## Examples regarding derivatives of linear function

(1) Is the function f(x) = (x + 2)^{2} linear. If so, find the derivation:

Evaluating f(x) = x^{2} + 2x + 4. Here highest power of x is 2 which is greater than 1. Hence the function is non-linear.

(2) Find the derivative of a line f(x) = 4x + 3:

Here f(x) = 4x + 3. Comparing with f(x) = mx + c, we have the slope m = 4 and we know that derivative of linear function of line is its slope. Hence f^{'}'(x) = 4.

Engineer
San Francisco, USA
"If you're at school or you just deal with mathematics, you need to use Studygeek.org. This thing is really helpful." |
Math Teacher
New-York, USA
"I will recommend Studygeek to students, who have some chalenges in mathematics. This Site has bunch of great lessons and examples. " |
Student, Designer
Philadelphia, USA
" I'm a geek, and I love this website. It really helped me during my math classes. Check it out) " |
Bookkeeper
Vancuver, Canada
"I use Studygeek.org a lot on a daily basis, helping my son with his geometry classes. Also, it has very cool math solver, which makes study process pretty fun" |