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 derivative of a linear function 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) = x2 + 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.

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