Derivative of the Linear Function

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

Let us suppose that the linear function is of the form $$y = mx + c$$, where $$m$$ is the slope and $$c$$ is Y-intercept, and this 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 differentiating separately we get
\[\begin{gathered} \Rightarrow \frac{{dy}}{{dx}} = \frac{d}{{dx}}mx + \frac{d}{{dx}}c \\ \Rightarrow \frac{{dy}}{{dx}} = m\frac{d}{{dx}}x + \frac{d}{{dx}}c \\ \end{gathered} \]

Use the following formulae $$\frac{d}{{dx}}\left( x \right) = 1$$ and $$\frac{d}{{dx}}\left( c \right) = 0$$ to 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 can conclude that the derivative of the given function is the slope of the function.


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

We have the given function as
\[y = 9\]

Differentiating with respect to variable $$x$$, we get
\[\frac{{dy}}{{dx}} = \frac{d}{{dx}}9x + \frac{d}{{dx}}10\]

Now using the formulas $$\frac{d}{{dx}}\left( x \right) = 1$$and $$\frac{d}{{dx}}\left( c \right) = 0$$, we have
\[\begin{gathered} \frac{{dy}}{{dx}} = 9\left( 1 \right) + 0 \\ \Rightarrow \frac{d}{{dx}}\left( {9x + 10} \right) = 9 \\ \end{gathered} \]