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Derivatives

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Derivative

Derivative, in mathematics, is the rate of change of a function with respect to a variable.

Geometrically, the derivative of a function can be interpreted as the slope of the graph of the function or, more precisely, as the slope of the tangent line at a point ,i.e,

$Slope = \frac{Change in y}{Change in x} = \frac{Δy}{Δx}$

The derivative of function $f$ at point $x$ ,denoted by $f^{'} (x)$ is given by:

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

if the limit exists finetely.

Alternatively , we may also define the derivative of $f (x)$ at point $a$ where $a$ belogs to any open set $S$ as:

$f^{'} (a) = \lim_{x \to a} \frac{f (x) - f (a)}{x - a}$

From the MathApp below , the definition of derivative is explained that how the change in value of $h$ and change in value of $x$ will effect the slope of the curve. You can drag the parameters $x$ and $h$ to see how slope is changing. Also the slope is calculated in the MathApp.