The derivative of $f(x)=mx+b$.

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Lesson Parent: 
Differentiation II: The Gory Details of Calculating Derivatives
Problem: 

Use the function limit definition of a derivative to determine the slope of a straight line.

Answer: 

$f'(x)=m$

SAMPLE SOLUTION

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Solution: 

We know that the straight line is defined by the equation <Sign in to see all the formulas> where $m$ is the slope of the function and $b$ is the $y$-intercept. Since $m$ is the slope of the straight line and the derivative is the slope of the tangent line, we therefore expect that <Sign in to see all the formulas>.

Let's construct the function <Sign in to see all the formulas> and then take the limit at $h$ goes to zero. First, <Sign in to see all the formulas>. Now we have
\begin{eqnarray*}
f'(x)&=&\lim_{h\to 0}\frac{f(x+h)-f(x)}{h},\\
&=&\lim_{h\to 0}\frac{m\,x+m\,h+b-m\,x-b}{h},\\
\mbox{and}&&\\
&=&\lim_{h\to 0}\frac{m\,h}{h},\\
\end{eqnarray*}

Now we know that

<Sign in to see all the formulas>



and that a constant times a limit equals the limit of the constant times the function. Therefore

<Sign in to see all the formulas>

Therefore, <Sign in to see all the formulas> and we obtained the known slope of the function.