13. Limits and Derivatives

13 Limits and Derivatives class 11 maths

LIMITS AND DERIVATIVES CLASS 11 MATHS

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_{x \to a} {f(x)} and f(a) may not be same (In fact, one may be defined and not the other one).

For functions f and g the following holds:
\lim_{x\to a} [f(x) \pm g(x)] = \lim_{x \to a}f(x) \pm \lim_{x \to a}g(x)
\lim_{x\to a} [f(x) \times g(x)] = \lim_{x \to a}f(x) \times \lim_{x \to a}g(x)
\lim_{x\to a} \frac{f(x)} {g(x)} = \frac {\lim_{x \to a} f(x)} {\lim_{x \to a} g(x)}

Following are some of the standard limits
\lim_{x\to a} \frac {x^n-a^n}{x-a} = n a^{n-1}
\lim_{x\to 0} \frac {sin(x)}{x} = 1
\lim_{x\to 0} \frac {1-cos(x)}{x} = 0

The derivative of a function f at a is defined by
f'(a) =  \lim_{h\to 0} \frac {f(a+h)-f(a)}{h} 

Derivative of a function f at any point x is defined by
f'(x) = \frac{d}{dx} f(x) =  \lim_{h\to 0} \frac {f(x+h)-f(x)}{h} 

For functions u and v the following holds:
(u \pm v)' = u' \pm v'
(uv)' = uv' + vu'
\left(\frac{u}{v} \right)'=\frac{u'v-uv'}{v^2} provided all are defined.

Following are some of the standard derivatives.
\frac{d}{dx} (x^n) = n x^{n-1}
\frac{d}{dx} (sin x) = cos x
\frac{d}{dx} (cos x) = -sin x

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Lecture - 1

  • Meaning of Limits
  • Different methods of finding limits

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Lecture - 2

  • NCERT Example 1, 2
  • NCERT Exercise 13.1
  • Identities in limits

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Lecture - 3

  • Logarithmic Functions
  • Exponential Functions
  • NCERT Supplementary Exercise
  • Few complex question

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Lecture - 4

  • Limits on Double definition functions
  • NCERT Exercise 13.1

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Lecture - 5

  • Meaning of Derivatives
  • First Principle Derivation

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Lecture - 6

  • First Principle questions
  • Base of Derivatives

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Lecture - 7

  • NCERT Example 19, 20
  • NCERT Exercise 13.2 Q4
  • Miscellaneous Exercise Q1

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Lecture - 8

  • Direct Derivative
  • Product Rule
  • Quotient Rule

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Lecture - 9

  • Direct Derivatives
  • NCERT Exercise 13.2

Videos & PDFs

Lecture - 10

  • NCERT Miscellaneous Exercise Q2 to Q11, Q15 to Q18, Q20, Q22 to Q29

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Lecture - 11

  • Chain Rule
  • NCERT Miscellaneous Exercise

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