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Differentiation

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).

Interpretation of the Derivative as a Rate of Change

Suppose that 'y' is a quantity that depends on another quantity 'x'. Thus, 'y' is a function of 'x' and we write y = f(x). If 'x' changes from x1 to x2, then the change in 'x', also called the increment of 'x', is: Δx = x2 – x1 and the corresponding change in 'y' is: Δy = f(x2) – f(x1). The difference quotient: is called the average rate of change of 'y' w. r. t. 'x' over the interval [x1, x2] and can be interpreted as the slope of the secant line PQ in below figure.

Consider the average rate of change over smaller and smaller intervals by letting x2 approach x1 and therefore letting Δx approach 0. The limit of these average rates of change is called the instantaneous rate of change of 'y' with respect to 'x' at x = x1, which is interpreted as the slope of the tangent to the curve y = f(x) at P(x1, f(x1)). Thus, instantaneous rate of change = ⋅⋅⋅⋅⋅⋅ (i).

From equation (i), we recognize this limit as being the derivative of f(x) at x1, that is, f '(x1). Therefore, the derivative is the instantaneous rate of change of y = f(x) with respect to 'x' when x = c.

  • Ex: The position of a particle is given by the equation of motion s = f(t) = t2 – 2t + 2, where 's' is measured in meters and 't' in seconds. Find the velocity after 3 seconds.

    Sol:
Rules for Differentiation

Constant rule: If f(x) is the function with the constant value 'c', then the derivative of f(x) is: [f(x)] = [c] = 0. Thus, the constant rule states that the derivative of any constant function is '0'.

Power rule: If f(x) = xn, where 'n' is any real number, then the derivative of f(x) is: [f(x)] = [xn] = n.xn – 1. Thus, the power rule states that the derivative of x raised to a power is the power times x raised to a power one less.

  • Ex: The cost of producing the 'x' ounces of gold from a new gold mines is given as: C = f(x), where f(x) = x4. Find (i) f '(x) (ii) f '(2).
    Sol: Given f(x) = x4 [f(x)] = [x4] = 4.x4 - 1 = 4x3
           f '(x) = 4x3 ⇒ f '(2) = 4[23] = 32

Sum rule: If 'u', 'v' are two differentiable functions of 'x', then (u + v) = + . Thus, the sum rule states that the derivative of the sum of two functions is the sum of the derivatives of the functions.

  • Ex: Find (x2 + 6x).

    Sol: (x2 + 6x) = (x2) + (6x) = 2.x2 – 1 + 6 . (x) = 2x + 6.

Difference rule: If 'u', 'v' are two differentiable functions of 'x', then (u – v) = . Thus, the difference rule states that the derivative of the difference of two functions is the difference of the derivatives of the functions.

  • Ex: Find (2x3 – 5).

    Sol: (2x3 – 5) = (2x3) – (5) = (2)(3)x3 – 1 – 0 = 6x2.
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