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The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of the function of $f$ is $f'$, pronounced "$f$ prime." For instance, if $f(x) = x^2$ is the squaring function, then $f'(x) = 2x$ is its derivative, the doubling function. If the input of the function represents time, then the derivative represents change with respect to time. For example, if f is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of f is how the position is changing in time, that is, it is the velocity of the ball. If a function is linear (that is, if the graph of the function is a straight line), then the function can be written as $y = mx + b$, where $x$ is the independent variable, $y$ is the dependent variable, $b$ is the $y$-intercept, and: $$m= \frac{\text{rise}}{\text{run}}= \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}.$$ This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in $y$ divided by the change in $x$ varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let $f$ be a function, and fix a point $a$ in the domain of $f$. $(a, f(a))$ is a point on the graph of the function. If $h$ is a number close to zero, then $a + h$ is a number close to $a$. Therefore $(a + h, f(a + h))$ is close to $(a, f(a))$. The slope between these two points is $$m = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h}.$$ This expression is called a difference quotient. A line through two points on a curve is called a secant line, so $m$ is the slope of the secant line between $(a, f(a))$ and $(a + h, f(a + h))$. The secant line is only an approximation to the behavior of the function at the point $a$ because it does not account for what happens between $a$ and $a + h$. It is not possible to discover the behavior at $a$ by setting $h$ to zero because this would require dividing by zero, which is impossible. The derivative is defined by taking the limit as $h$ tends to zero, meaning that it considers the behavior of $f$ for all small values of $h$ and extracts a consistent value for the case when $h$ equals zero: $$\lim_{h \to 0}{f(a+h) - f(a)\over{h}}.$$ Geometrically, the derivative is the slope of the tangent line to the graph of $f$ at $a$. The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function $f$. Here is a particular example, the derivative of the squaring function at the input $3$. Let $f(x) = x^2$ be the squaring function. $$ \begin{align}f'(3) &=\lim_{h \to 0}{(3+h)^2 - 3^2\over{h}} \\ &=\lim_{h \to 0}{9 + 6h + h^2 - 9\over{h}} \\ &=\lim_{h \to 0}{6h + h^2\over{h}} \\ &=\lim_{h \to 0} (6 + h) \\ &= 6. \end{align} $$ The slope of tangent line to the squaring function at the point $(3,9)$ is $6$, that is to say, it is going up six times as fast as it is going to the right. The limit process just described can be performed for any point in the domain of the squaring function. This defines the derivative function of the squaring function, or just the derivative of the squaring function for short. A similar computation to the one above shows that the derivative of the squaring function is the doubling function. [from http://en.wikipedia.org/wiki/Calculus]
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