Limits and Derivatives | Class 11 Mathematics Notes

Rules of Differentiation

This section introduces the basic rules for finding derivatives of functions without resorting to the limit definition every time. The sum rule states that the derivative of a sum is the sum of the derivatives: (u + v)' = u' + v'. The product rule states that the derivative of a product is given by (uv)' = u'v + uv'. The quotient rule states that the derivative of a quotient is (u/v)' = (u'v - uv') / v², provided v ≠ 0. The chain rule, though not covered in detail here, is essential for composite functions. These rules allow efficient computation of derivatives for polynomials, trigonometric functions, and other standard functions. The section also discusses the derivative of constant functions, which is zero, and the derivative of the identity function f(x) = x, which is 1. Examples demonstrate the application of these rules to find derivatives of various functions.

📊 Diagram: Table on page 38 (2×3) showing differentiation using quotient rule.

🧪 Activity: Practice exercises applying differentiation rules to various functions.

🔗 Connection: Leads to derivatives of standard functions and more complex examples.

Table on page 37 (1×2)

| x+cosx (ii) We use quotient rule on the function wherever it is defined. tanx (x+cos x) ′tanx−(x+cos x)(tan x)′ h ′( x) = (tan x)2 (1−sin x)tanx−(x+cos x)sec2 x = (tan x)2 Miscellaneous Exercise on Chapter 12 1. Find the derivative of the following functions from first principle: π (i) −x (ii) (−x )−1 (iii) sin (x + 1) (iv) cos (x – ) 8 Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): r  2. (x + a) 3. (px + q)  +s 4. (ax+b)(cx+d)2 x  1 1+ ax+b 1 x 5. 6. 7. cx+d 1 ax2 +bx+c 1− x ax+b px2 +qx+r a b 8. 9. 10. − +cosx px2 +qx+r ax+b x4 x2 | |

Table on page 3 (1×2)

| Table 12.3 t 4 3 2.5 2.2 2.1 2.05 2.01 2 v 29.4 24.5 22.05 20.58 20.09 19.845 19.649 Here again we note that if we take smaller time intervals starting at t = 2, we get better idea of the velocity at t = 2. In the first set of computations, what we have done is to find average velocities in increasing time intervals ending at t = 2 and then hope that nothing dramatic happens just before t = 2. In the second set of computations, we have found the average velocities decreasing in time intervals ending at t = 2 and then hope that nothing dramatic happens just after t = 2. Purely on the physical grounds, both these sequences of average velocities must approach a common limit. We can safely conclude that the velocity of the body at t = 2 is between 19.551m/s and 19.649 m/s. Technically, we say that the instantaneous velocity at t = 2 is between 19.551 m/s and 19.649 m/s. As is well-known, velocity is the rate of change of displacement. Hence what we have accomplished is the following. From the given data of distance covered at various time instants we have estimated the rate of change of the distance at a given instant of time. We say that the derivative of the distance function s = 4.9t2 at t = 2 is between 19.551 and 19.649. An alternate way of viewing this limiting process is shown in Fig 12.1. This is a plot of distance s of the body from the top of the cliff versus the time t elapsed. In the limit as the sequence of time intervals h , h , ..., approaches | |

| 1 2 zero, the sequence of average velocities approaches the same limit as does the | |

Table on page 4 (1×2)

| v(t) of a body at time t = 2 is equal to the slope of the tangent of the curve s = 4.9t2 at t = 2. 12.3 Limits The above discussion clearly points towards the fact that we need to understand limit- ing process in greater clarity. We study a few illustrative examples to gain some famil- iarity with the concept of limits. Consider the function f(x) = x2. Observe that as x takes values very close to 0, the value of f(x) also moves towards 0 (See Fig 2.10 Chapter 2). We say lim f ( x)=0 x→0 (to be read as limit of f (x) as x tends to zero equals zero). The limit of f (x) as x tends to zero is to be thought of as the value f (x) should assume at x = 0. In general as x → a, f (x) → l, then l is called limit of the function f (x) which is symbolically written as lim f ( x)= l. x→a Consider the following function g(x) = |x|, x ≠0. Observe that g(0) is not defined. Computing the value of g(x) for values of x very near to 0, we see that the value of g(x) moves lim towards 0. So, g(x) = 0. This is intuitively x→0 clear from the graph of y = |x| for x ≠0. (See Fig 2.13, Chapter 2). Consider the following function. x2 −4 h ( x)= , x≠2. x−2 Compute the value of h(x) for values of x very near to 2 (but not at 2). Convince yourself | |

Table on page 7 (1×2)

| lim f ( x)= 1. x→1+ Hence, it is likely that the left hand limit of f(x) and the right hand limit of f(x) are both equal to 1. Thus, lim f ( x)= lim f ( x)=lim f ( x)= 1. x→1− x→1+ x→1 This conclusion about the limit being equal to 1 is somewhat strengthened by seeing the graph of this function which is given in Fig 2.11, Chapter 2. In this figure, we note that as x approaches 1 from either right or left, the graph of the function f(x) = x3 approaches the point (1, 1). We observe, again, that the value of the function at x = 1 also happens to be equal to 1. Illustration 3 Consider the function f(x) = 3x. Let us try to find the limit of this function at x = 2. The following Table 12.6 is now self-explanatory. Table 12.6 x 1.9 1.95 1.99 1.999 2.001 2.01 2.1 f(x) 5.7 5.85 5.97 5.997 6.003 6.03 6.3 As before we observe that as x approaches 2 from either left or right, the value of f(x) seem to approach 6. We record this as lim f ( x)= lim f ( x)=lim f ( x)=6 x→2− x→2+ x→2 Its graph shown in Fig 12.4 strengthens this fact. Here again we note that the value of the function at x = 2 coincides with the limit at x = 2. | |

Table on page 11 (2×3)

| lim f ( x)=2 x→0+ Since the left and right hand limits at 0 do not coincide, we say that the limit of the function at 0 does not exist. | | |

| Graph of this function is given in the Fig12.6. Here, we remark that the value of the function at x = 0 is well defined and is, indeed, equal to 0, but the limit of the function at x = 0 is not even defined. Fig 12.6 Illustration 10 As a final illustration, we find lim f ( x) , x→1 where x+2 x≠1 f ( x)=   0 x=1 Table 12.12 x 0.9 0.99 0.999 1.001 1.01 1.1 f(x) 2.9 2.99 2.999 3.001 3.01 3.1 As usual we tabulate the values of f(x) for x near 1. From the values of f(x) for x less than 1, it seems that the function should take value 3 at x = 1., i.e., lim f ( x)=3. x→1− Similarly, the value of f(x) should be 3 as dic- tated by values of f(x) at x greater than 1. i.e. lim f ( x)=3. x→1+ But then the left and right hand limits coin- cide and hence | | | | lim f ( x)= lim f ( x)=lim f ( x)=3. x→1− x→1+ x→1 | | |

Table on page 38 (2×3)

| 23. (x2 +1) cosx 24. (ax2 +sinx)(p+qcosx)  π 4x+5sinx x2 cos  25. (x+cosx) (x−tanx) 26. 27.  4 3x+7cosx sinx x x 28. 29. (x+secx) (x−tanx) 30. 1+tanx sinnx Summary ® The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit. ® Limit of a function at a point is the common value of the left and right hand limits, if they coincide. ® For a function f and a real number a, lim f(x) and f (a) may not be same (In x→a fact, one may be defined and not the other one). ® For functions f and g the following holds: lim[f ( x)±g ( x)]=lim f ( x)±lim g ( x) x→a x→a x→a lim[f ( x ). g ( x)]=lim f ( x).limg ( x) x→a x→a x→a lim f ( x)  f ( x) lim =x→a x→ag ( x) lim g ( x) x→a ® | | |

| | Summary ® The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit. ® Limit of a function at a point is the common value of the left and right hand limits, if they coincide. ® For a function f and a real number a, lim f(x) and f (a) may not be same (In x→a fact, one may be defined and not the other one). ® For functions f and g the following holds: lim[f ( x)±g ( x)]=lim f ( x)±lim g ( x) x→a x→a x→a lim[f ( x ). g ( x)]=lim f ( x).limg ( x) x→a x→a x→a lim f ( x)  f ( x) lim =x→a x→ag ( x) lim g ( x) x→a ® | | | | Following are some of the standard limits xn −an lim =nan−1 x→a x−a | |