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
f′(x)=limh→0f(x+h)−f(x)h,=limh→0mx+mh+b−mx−bh,and=limh→0mhh,

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.