Lear UNE Online Concept of Differentiation An Overview of Differentiation In Maths, Differentiation in Class 11 is one of the most important topics both academically and in terms of marks weightage. The concept of differentiation refers to the method of finding the derivative of a function. It is the process of determining the rate of change in function on the basis of its variables. The opposite of differentiation is known as anti-differentiation. Suppose, we have two variables x and y. Then, the rate of change of x with respect to y is denoted as dy/dx. The general expression of the derivative of a function is f'(x)= dy/dx where y= f(x) is any function. (Image will be uploaded soon) What is Differentiation in Mathematics? Differentiation in Mathematics is defined as a derivative of a function in terms of an independent variable. Let f(x) be a function of x and y be another variable. Here, the rate of change of y per unit change in x is denoted by dy/dx. If the function f(x) goes through an infinitesimal change of h near to any point x, the function is defined as, Lim f( +h) ~ f00/ h tends to 0.Lear UNE Online What is Differentiation in Physics? Differentiation in physics is the same as differentiation in Mathematics. The concepts from differentiation in Maths are used in physics too. Some Important Formulas in Differentiation Some important differentiation formulas in Class 11 are given below. We have to consider f(x) as a function and f(x) as the derivative of the function: 1. If f(x) = tan(x) then f(x) = sec2x. 2. If f(x) = cos(x) then f(x) = ~sin x. 3. If f(x) = sin(x) then f(x) = cos x. 4. If f(x) = In(x) then f(x) = 1/x. 5. If f(x) = e* then f(x) = eX. 6. If f(x) = x" then f'(x) = nx! where n is any fraction or integer. 7. If f(x) = k then f'(x) = 0 and here k is a constant. Rules of Differentiation The main differentiation rules that need to be followed are given below: 1. Product Rule 2. Sum and Difference Rule 3. Chain Rule 4. Quotient Rule Product Rule - According to the product rule, if the function f(x) is the product of two functions suppose a(x) and b(x), then the derivative of that function is: IF F(X) = a(x) * b(x) then,Lear UNE Online F(x) = a'(x) * B(x) + a(x) *b'() Sum and Difference Rule — According to the sum and difference rule, if the function f(x) is the sum or difference of two functions suppose a(x) and b(x), then the derivative of the function is as follows: If £0) = a(x) + b(W) then, (x) = al(x)+ bx) If £(%) = a(x) ~ b(®) then, F(x) = a(x) -b(%) Chain Rule - If a function y= f(x) = g(u) and if u = h(x), then according to the chain rule for differentiation, dy/dx = dy/du * du/dx This rule is very important in the method of substitution during differentiation of composite functions. Quotient Rule ~ If the function f(x) is the quotient of two functions i.e. a(x)/b(X), then according to quotient rule, the derivative of the function is as follows: IF F(X) = a(x)/b(%)Lear UNE Online then, f(x) = a(x) * b(x) ~ a(x) * (x) / (b(x))? As a result, for individual functions composed of combinations of these classes, the theory provides the following basic rules for differentiating either the sum, product, or quotient of two functions f(x) and g(x), whose derivative is known (where a and b are constant). D(af + bg) = aDf + bDg (sum); D(fg) = fDg + gDf (product); and D(f/g) = (gDf - fDg)/g2 (quotients). Other basic rules can be applied to composite functions, including the chain rule. By taking the value of g(x) and f(x) as inputs to the composite function f(g(x)), where f(x) = sin x2, g(f(x)) = (sin x2) , all calculated for a given value of x; for instance, if f(x) = sin x and g(x) = x2, then g(f(x)) = (sin x2), The chain rule provides that the derivative of a composite function is equal to the product of the derivatives of the component functions. So, D(f(g(x)) = Df(g(x)) - Dg(x). It is necessary to first find the derivative of Df(x), and x, as necessary, is then replaced by the function g(x). This is the first factor on the right. Using the rule shown above, we get D(sin x2) = D sin(x2) - D(x) = (cos X2) - 2x. The chain rule takes on the more memorable “symbolic cancellation” form in the notation of German mathematician Gottfried Wilhelm Leibniz, which uses d/dx in place of D to permit differentiation according to variables, such as: d(f(g(x)))/dx = df/dg - dg/dx. Solved Example 1. Differentiate f(x) = 9x3 - 6x + 5 with respect to x. Solution: Here, f(x) = 9x3 - 6x + 5. Differentiating both sides wart. x, we get, F(x) =(3)(9)x2 - 6F(x) = 27x2 -6 HiDivyam, Best courses for you Full syllabus LIVE courses Starting fom 83,667/month Books Starting from 283 LIVE Masterclass Show more courses V Vedaniti, Lear NEOnlne One-to-one LIVE classes Stating fom 21,800 Test series Stating trom #57Vedaniti, Lear NEOnlne Sree enact eee On ee or Recars Book your Free Demo session Get a flavour of LIVE classes here at Vedantu Vedantu Improvement Promise We promise improvement in marks or get your fees back. T&C Apply*