Derivative of Linear Function

In this tutorial we discuss the derivative of linear function or derivative of straight line equation in the form of slope intercept.

Let us suppose that linear function of the form y = mx + c where m is slope and c is Y-intercept represents the equation of straight line.

y = mx + c

Differentiating the given linear function with respect to x, we get

 \Rightarrow \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left( {mx + c} \right)

In the given function we have two values, now differentiate separately we get

\begin{gathered} \Rightarrow \frac{{dy}}{{dx}} = \frac{d}{{dx}}mx + \frac{d}{{dx}}c \\ \Rightarrow \frac{{dy}}{{dx}} = m\frac{d}{{dx}}x + \frac{d}{{dx}}c \\ \end{gathered}

Using the following formulae \frac{d}{{dx}}\left( x \right) = 1 and \frac{d}{{dx}}\left( c \right) = 0to evaluate given function as

\begin{gathered} \Rightarrow \frac{{dy}}{{dx}} = m\left( 1 \right) + \left( 0 \right) \\ \Rightarrow \frac{{dy}}{{dx}} = m \\ \end{gathered}

We concluded that the derivative of given function is the slope of function.

Example: Find the derivative of y = f\left( x \right) = 9x + 10

We have the given function as

y = 9

Differentiation with respect to variable x, we get

\frac{{dy}}{{dx}} = \frac{d}{{dx}}9x + \frac{d}{{dx}}10

Now using the formulas \frac{d}{{dx}}\left( x \right) = 1and \frac{d}{{dx}}\left( c \right) = 0, we have

\begin{gathered} \frac{{dy}}{{dx}} = 9\left( 1 \right) + 0 \\ \Rightarrow \frac{d}{{dx}}\left( {9x + 10} \right) = 9 \\ \end{gathered}