How do I use the limit definition of derivative to find f'(x) for f(x)=mx+b?

Question

How do I use the limit definition of derivative to find f&#039;(x) for f(x)=mx+b?

glanzerjbdo · Accepted Answer

Remember that the limit definition of the derivative goes like this:

f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}

So, for the posted function, we have

f^{'} (x) = lim_{h \to 0} \frac{m (x + h) + b - [m x + b]}{h}

By multiplying out the numerator,

= lim_{h \to 0} \frac{m x + m h + b - m x - b}{h}

By cancelling out mx's and b's,

= lim_{h \to 0} \frac{m h}{h}

By cancellng out h's,

= lim_{h \to 0} m = m

Hence, f'(x)=m.
The answer above makes sense since the derivative tells us about the slope of the tangent line to the graph of f, and the slope of the linear function (its graph is a line) is m.

How do I use the limit definition of

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