Step By Step Calculus » 10.1 - Definition of Derivative

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Synopsis
We define the derivative of a function f:D_{f} \rightarrow E_{f}f:D_{f} \rightarrow E_{f} at a point a\in D_{f} a\in D_{f} to be the slope of the tangent line to the graph of ff at the point (a, f(a))(a, f(a)). The derivative is therefore defined as the limit of the slope of the secant line through the points (x, f(x))(x, f(x)) and (a. f(a))(a. f(a)) as xx approaches aa:
\displaystyle f'(a) = \lim\limits_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}
\displaystyle f'(a) = \lim\limits_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}
It can be also written as the following:
\displaystyle f^\prime(a)=\lim\limits_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}
\displaystyle f^\prime(a)=\lim\limits_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}
where h \rightarrow 0h \rightarrow 0 from both sides of 00. The limit which defines the derivative can also be written
\displaystyle f^\prime(a)=\lim\limits_{h\rightarrow 0}\frac{f(a)-f(a-h)}{h}
\displaystyle f^\prime(a)=\lim\limits_{h\rightarrow 0}\frac{f(a)-f(a-h)}{h}
where as always h \rightarrow 0h \rightarrow 0 from both sides.
It is also possible to define the derivative from the left (f^\prime_-(a)f^\prime_-(a)) and the right (f^\prime_+(a)f^\prime_+(a)) by using the appropriate one-sided limit .
If the derivative of f(x)f(x) is defined at some x=ax=a, then f(x)f(x) is said to be differentiable at aa. There are several ways for differentiability to fail at aa, including (but not limited to!) the following:
  • A discontinuity at aa.
  • A corner at aa.
  • A vertical tangent at aa.
Given y=f(x)y=f(x), the following are all understood to denote the derivative: \displaystyle y^\prime\displaystyle y^\prime, \displaystyle f^\prime(x)\displaystyle f^\prime(x), \displaystyle \frac{dy}{dx}\displaystyle \frac{dy}{dx}, \displaystyle \frac{df}{dx}\displaystyle \frac{df}{dx}, \displaystyle \frac{d}{dx}f(x)\displaystyle \frac{d}{dx}f(x), DfDf. We avoid the last notation, DfDf, because in this text we often use DD to denote the domain of a function.
The derivative of an even function is odd and vice versa.