NCERT Solutions for Class 11 Maths Chapter 13

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 Class: 11 Subject: Maths Chapter 13: Limits and Derivatives

NCERT Solutions for Class 11 Maths Chapter 13

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Class 11 Maths Chapter 13 Limits and Derivatives Sols

Important Terms Related to Limits & Derivatives

• Limits: Meaning of x approaches (x->a): Let x be a variable and ‘a’ be a fixed number. If x assumes a succession of values nearer and nearer to ‘a’, it is said to approach ‘a’. We mean that x assumes successively values [neither greater nor less than a] there is no end to this process of coming nearer and nearer to ‘a’. In this process (x – a) becomes smaller and smaller and can be made as small as we like. We express this by saying that |x – a| become smaller than δ, where δ is an arbitrary number which may be as very small as we please.

• Definition of x ->a: Let x be a variable and ‘a’ be a fixed number. Given a positive number δ however small it may be, if x assumes values such that 0 < |x – a| < δ, then we say x tends to ‘a’ and write x ->a.
• Concept of Limit: Let any function f(x) = y, is not defined at x = a, then the value of function at x = a is f(a), is meaningless. In this case the value of x can be taken nearer to a. Then the value of f(x) is calculated at that nearer value. Hence, the value of f(x) nearer to a. This limiting value of f(x) is known as limit of x.

lim (x→3):⁡〖x+3〗
= 3 + 3
= 6

Find the derivative of x^n+ax^(n-1)+a^2 x^(n-2)+⋯+a^(n-1) x+a^n for some fixed real number a.

Let f(x)=x^n+ax^(n-1)+a^2 x^(n-2)+⋯+a^(n-1) x+a^n
∴f^’ (x)=d/dx(x^n+ax^(n-1)+a^2 x^(n-2)+⋯+a^(n-1) x+a^n)
=d/dx (x^n )+a d/dx (x^(n-1) )+a^2 d/dx (x^(n-2) )+⋯+(a^(n-1 ) d)/dx (x)+a^n d/dx (1)
On using theorem d/dx x^n=nx^(n-1) ,we obtain
f^’ (x)=nx^(n-1)+a(n-1) x^(n-2)+a^2 (n-2) x^(n-3)+⋯+a^(n-1)+a^n (0)
=nx^(n-1)+a(n-1) x^(n-2)+a^2 (n-2) x^(n-3)+⋯+a^(n-1)

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): (px+q)(r/x+s).

Let f(x)=(px+q)(r/x+s)
f'(x)=(px+q) (r/x+s)’+(r/x+s) (px+q)’
=(px+q) (rx^(-1)+s)’+(r/x+s)(p)
=(px+q)(-rx^(-2) )+(r/x+s)p
=(px+q)((-r)/x^2 )+(r/x+s)p
=(-pr)/x-qr/x^2 +pr/x+ps
=ps-qr/x^2