Limits and Derivatives

Standard Simplifications

Order and simplification are the first steps to the mastery of a subject”. This quote by Thomas Mann encapsulates the concept we discuss here. In this discussion, we will see some standard simplifications used in derivatives. As we have already studied the concept of limits and derivatives. We have also studied their algebraic proofs using detailed analysis to understand the concept in detail.

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Standard Simplification: Derivatives

We know that for a function f(x), the derivative function is defined as

f ‘(x) = limh→0[f(x+h)-f(x)]/h

We have studied its rigorous proofs in the previous article with examples. Now we shall look at some rules to applications of derivatives to functions and we shall also see the conditions that need to be achieved in order to apply these rules to get correct results.


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Algebra of Derivatives

Since derivatives are derived from limits, their rule set too is similar in function to limits.they can be seen as below. Let ‘f’ and ‘g’ be 2 functions such that their derivatives lie in a common domain. Then

1. The derivative of the sum of two functions is the sum of the derivatives of both the functions.

d(f(x) + g(x))/dx = d(f(x))/dx + d(g(x))/dx

2. Similarly, the derivative of the difference of two functions is the difference of derivatives of these functions.

d(f(x)-g(x))/dx= d(f(x))/dx – d(g(x))/dx

Leibnitz Rule

Let u=f(x) and v=g(x) be two functions, then the product rule for their derivatives is

(uv)’ = u’v + uv’

Similarly, the quotient rule for the derivative of their division is

(u/v)’= (u’v-uv’)/v2

3. The derivative of the product of two functions is given by the product rule

d(f(x).g(x))/dx= d(f(x))/dx.g(x) + f(x).d(g(x))/dx

4. The derivative of the division of two functions is given by the quotient rule

d(f(x)/g(x))/dx= (d(f(x))/dx.g(x) + f(x).d(g(x))/dx)/(g(x)2)

Using all the above theorems we can now look at some standard results that have been derived prior hand to ensure easy calculations of derivatives of complex functions.


The derivative of f(x)=xn  is nxn-1 for any positive integer

Proof: By definition of the derivative function we have

f ‘(x) = limh→0f(x+h)-f(x)/h

= limh→0((x+h)n-xn)/h

Binomial theorem states that (x+h)n = (nC0)xn.h + (nC1)xn-1.h+…. Thus,

f ‘(x)=limh→0(nxn-1+….+hn-1)=nxn-1

Similarly, we also have standard derivatives of many other functions. The scope of this discussion is keeping in mind the syllabus. At the same time, we also aim to keep it as simple as possible.

Therefore, we present a few such functions and their standard derivatives. These derivatives can be used in the calculation of complex functions. Care must be taken of specified limits and conditions.

  • If f(x) = a (a= real number) then, f ‘(x) = 0
  • If  f(x) = 1/x then, f ‘(x) = -1/x2
  • When f(x) = tanx then, f ‘(x) = sec2x
  • If f(x) = sinx then, f ‘(x) = cosx

Solved Example for You

Question 1: Find the derivative of sin2x.

Answer: Let f(x)=sin2x

Using Leibnitz rule, f(x) = (sinx.sinx)

From the product rule, f ‘ (x) = (sinx’. sinx) + (sinx.sinx’)

Using standard simplification, f ‘ (x) = (cosx.sinx) + (sinx.cosx)

= 2 sinx.cosx = sin2x

Question 2: What is a simplification in math?

Answer: In maths, simplification is the method of replacing a mathematical expression by an equivalent one. This equivalent one is usually simpler and shorter. For instance, simplifying a fraction to an irreducible fraction.

Question 3: What is the definition of derivative formula?

Answer: Basically, the essence of calculus is derivative. Thus, a derivative is the immediate rate of change of a function of regarding one of its variables. Thus, it is equal to finding out the slope of the tangent line to the function at a point.

Question 4: Are limits and derivatives the same?

Answer: Basically, a derivative is merely a specific kind of limit. The derivative is said to be the slope of a function at some point on the function. Thus, the limit is the best guess at where the function will ultimately end up when it is approaching a specific number.

Question 5: What is product rule for derivatives?

Answer: The product rule means that if are multiplying two ‘parts’ of the function together. Similarly, the chain rule is when and if they compose. For instance, to find out the derivative of f(x) = x² sin(x), you make use of the product rule. Similarly, to find out the derivative of g(x) = sin(x²) you make use of the chain rule.

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