Definition of the Derivative

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One tricky part of calculus is the definition of the derivative. It’s tricky because the definition is kind of weird, you see it once, and then you don’t really use it again. Since it’s easier to calculate derivatives by using the rules you learn later, people often forget how the derivative is defined.

So let’s take a look! The derivative is the slope of the tangent line, so we might start by finding the slope of an appropriate secant line. If we have a function f, a value x, and a number h close to zero, the rise of the function between x and x+h is f(x+h)-f(x) and the run is h. Therefore the slope is

\displaystyle \frac{f(x+h)-f(x)}{h}.

Taking the limit as h goes to zero, we get the slope of the tangent line at h, f'(x):

\displaystyle f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}

We can use this to find the derivatives of simple functions, like g(x)=2x. Here \lim_{h\rightarrow 0}\frac{g(x+h)-g(x)}{h} is equal to

\displaystyle \lim_{h\rightarrow 0}\frac{2(x+h)-2x}{h}

and so after some algebra we find that g'(x)=2 for any value of x. This definition can be used to figure out most of the derivative formulas (the power rule, product rule, quotient rule, etc.), so it’s useful to remember.