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Maths > Limits and Derivatives > First Principle of Differentiation
Limits and Derivatives

First Principle of Differentiation

Hello friends! We have already studied the concepts of limits and derivatives. We have also seen standard substitutions and the algebra of both these concepts. Therefore, we have a basic understanding of these concepts in Calculus. In this discussion, we will derive the concept of the first principle of differentiation.

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The First Principle of Differentiation

We will now derive and understand the concept of the first principle of a derivative. This principle is the basis of the concept of derivative in calculus. A thorough understanding of this concept will help students apply derivatives to various functions with ease.

We shall see that this concept is derived using algebraic methods. Nevertheless, the application of the concept remains irrelevant to its derivation methods. The main purpose of this discussion is to ensure that students have a clear idea of the concept of derivative.

Browse more Topics under Limits And Derivatives

Definition

Let f(x) be a real function in its domain. A function defined such that

limx->0[f(x+h)-f(x)]/h

if it exists is said to be derivative of the function f(x). This is known as the first principle of the derivative. The first principle of a derivative is also called the Delta Method. We shall now establish the algebraic proof of the principle

Proof: Let y = f(x) be a function and let A=(x , f(x)) and B= (x+h , f(x+h)) be close to each other on the graph of the function.Let the line f(x) intersect the line x + h at a point C. We know that

f ‘(x) =  limh->0f(x + h) – f(x) / h

Differentiation

 

In the triangle ABC, we can say that the ratio whose limit we are taking is equal to tan(BAC).  That is the slope of the chord AB. Thus in the limiting process as h->0, point B tends to A and so we get

limh->0 (f(x+h)-f(x)) / h = limB->A BC / AC

which means that the chord AB tends to the tangent at A for the curve y= f(x). Hence, we get f ‘(x)= tan α where α = angle made by the tangent at A on the X-axis. For a function F, we can define the derivative at every point. It the derivative exists at all points it becomes a new function known as f ‘.

Note that domain of the derivative is where the function f'(x) exists. Sometimes f ‘(x) is also denoted as d/dx (f(x)) or also as dy dx.

Solved Example for You on Differentiation

Question: Evaluate the derivative of f(x) = 1/x

Solution: We know that,

f ‘(x) = limh->0(f(x+h)-f(x))/h

so f ‘ (x)= limh->0 ((1/x+h)-1/x)/h

= limh->01/h. [(x-(x+h)]/(x(x+h))

= limh->01/h.(-h/(x.(x+h)))

= limh->0 -1/x.(x+h)

= -1/x2

This conludes our discussion on the topic of the first principle of differentiation.

Question: State the general differentiation formula?

Answer: There are six general differentiation formulas which are as follows:

  1. Power Rule: \((\frac{d}{dx}) (x^{n}) = nx^{n-1}\)
  2. Derivation of a constant, a: \( (\frac{d}{dx}) (a) = 0\)
  3. Derivation of a constant increased with function f: \( (\frac{d}{dx}) (a.f) = af’\)
  4. Sum Rule: \((\frac{d}{dx}) (f \pm g) = {f}’ \pm {g}’\)
  5. Product Rule: \((\frac{d}{dx}) (fg) = {fg}’ + {gf}’\)
  6. Quotient Rule: \(\frac{d}{dx} \left (\frac{f}{g} \right ) = \frac{{gf}’ – {fg}’}{g^{2}}\)

Question: What is the difference between differentiation and derivative?

Answer: Derivative of a function is the rate of change of the output value with respect to its input value. On the other hand, the differential is the actual change of a function. Most importantly, differentiation is the process of finding a derivative.

Question: What is the derivative of 0?

Answer: The derivative of 0 is 0 because in general, we have the following rule for finding the derivative of a constant function, f(x) = a.

Question: How to differentiate XY?

Answer: In ordered differentiation, the function starts with y and equals some terms with x in it. However, with inherent differentiation, the function y as part of the function such as in XY or both sides of an equation such as in XY or both sides of an equation.

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