Real life application of derivatives
There are several real life applications of derivatives. They are mostly used to make connections or calculate information like: growth of bacteria, population change, hours of daylight, radioactive decay, prices and inflation change, etc.
Derivative and its interpretations
Definition of the derivative of a function: The derivative of a function f(x) with respect to x, written as , is defined as: , if this limit exists. We read as: f prime of x.
Notation: There are many ways to denote the derivative of a function f(x) with respect to x. Besides , the most common notations are: Dxf(x) and f(x). If the function is in the form of y = f(x), then the most common notations for the derivative are: y', and Dxy.
Derivative of a function at a number: The derivative of a function f(x) with respect to x at a number x = c, written as , is defined as: , if this limit exists.
A function is said to be derivable or differentiable at x = c, if it has a derivative there. A function is said to be differentiable on an interval, if it is differentiable at every point of the interval.
Interpretation of the Derivative as the Slope of a Tangent
If a curve C has equation y = f(x) and we want to find the tangent to C at the point P(c, f(c)), then we consider a nearby point Q(c + h, f(c + h)), where h ≠ 0, and compute the slope of the secant line PQ: mPQ = . Then we let Q approach P along the curve C by letting h approach 0. If mPQ approaches a number m, then we define tangent t to be the line through P with slope m. This amounts to saying that the tangent line is the limiting position of the secant line PQ as Q approaches P.
The tangent line to the curve y = f(x) at the point P(c, f(c)) is the line through P with slope m = , provided that this limit exists. Since, by the definition of a derivative of the function, this is the same as the derivative . Thus, the tangent line to the curve y = f(x) at the point P(c, f(c)) is the line through (c, f(c)) whose slope is equal to , the derivative of f(x) at c.
If we use the point-slope form of the equation of a line, we can write an equation of the tangent line to the curve y = f(x) at the point (c, f(c)): y – f(c) = (x – c).