DIFFERENTIATION

DIFFERENTIATION

DIFFERENTIATION


DIFFERENTIATION

✤ Types of Differentation:→
1. Chain Rule: when we have given Composition of function we should use
     chain rule.
     Ex- y=f(g(x)) then dy/dx = f'(g(x)).d g(x)/dx
     Ex- y = sin(logx)
      dy/dx sin(log x).d/dx logx.
       = cos(log x). X
2. Differentation of implicit
  Function:→ Those functions in which x and y are mined with each
      other and we can't seperate them
     Ex x²+2xy+ y² = 1k etc.
    to diff them we should diff y again.
     Ex x²+y²=25
      2xy+y²=0
      Dy/dx = -x/y
3. Differentation of a function w.r.t another function:→
    let fm is diff. w.r.t g(n)
    then let
      u= f(x), v = g(x)
    find du/dx = and dv/dx
    Now du/dv = [du/dx] / [dv/dx]
4. Diff by logarithm -
   when we have given a function in the Power of another function we
   should use log
     Ex (Sinx) Cosx
      Let y = (Sinx) Case
      logy = log (Sinx) Case
      logy = Cosx.log(Sinx)
   diff both side.
     1/y X dy/dx = Cosx.1 log(Sinx) + log(Sinx)
 Differentation of Parametric functions : →
   Those function which are in the form of 3rd variable like
   x= a cost, y = 2asint etc.
   to diff. find dx/dt = dt/dy =
   dy/dx = (dy/dt) / (dx/dt)
✤ Second order derivatives:→
  when a function is differentiated 2 times then it is called 2nd order
  derivatives
    Ex - y=sinx
     1st order ---> dy = cosx
     2nd order ---> d2y/dx2 = -sin x
    Note:
     dy / dx = y1 or y1
     d2y / dx2= y’’ or y2
✤ differentation of explicit function:→
   Those function in which x and y are seperated
    Ex- y = 3x²+2x+9 etc.
✤ Some formula for diff
  1. d/dx logex = 1/x
  2. d/dx logax = 1/x loga , a>0
  3. d/dx ex = ex, 1
  4. d/dx ax = ax loga , a>0
  5. d/dx secx = secxtanx.
  6. d/dx cosecx = -cosecx cotx
  7. d/dx tanx = sec2x
  8. d/dx sec2x = 2sec2xtanx
  9. d/dx cotx = -cosec2x
  10. d/dx xn = n.xn-1
  11. addition Rule: →
    d/dx (f(x)+g(x)) = d/dx f(x)+d/dx g(x)
  12. subtraction Rule: →
    d/dx (f(x)-g(x)) = d/dx f(x)-d/dx g(x)
  13. scalar rule: →
    d/dx [k.f(x)] = k. d/dx f(x)
  14. Product rule: →
    d/dx (f(x)xg(x)) = f(x) d/dx g(x) + g(x) d/dx f(x)
  15. division Rule: →
    d/dx (f(x)/g(x)) = g(x)f'(x)-f(x)g'(x) / (g(x))2
✤ Some other formulas: →
  1. d/dx sin-1x = 1/√(1-x2)
  2. d/dx cos-1x = -1/√(1-x2)
  3. d/dx tan-1x = 1/1+x2
  4. d/dx cot-1x = -1/1+x2
  5. d/dx sec-1x = 1/(x√(x2-1))
  6. d/dx cosec-1x = -1/(x√(x2-1))
✤ Some formula for log :→
  1. log (mxn) = logm + logn
  2. log (m/n) = logm - logn
  3. log (mn) = n logm
  4. logm = 1
  5. emlogx = xm
  6. log1 = 0
  7. loge = N
❈ Continuity →
  Let f(x) is any real function. If it has no sudden breaking Point then it
  is continuous.
Note: Let f(x) is any function which is defined at x=a Such that
   lim x→a f(x) = f(a) this is the condition for Continuous.
Types of limit: →
   LHL (Left hand limit) = lim x→a f(x) = lim h→0 f(a-h)
   RHL = lim x→a+ f(x) = lim h→0 f(a+h)
  §. If LHL = RHL ⇒ limit exists.
  §. If RHL ≠ LHL, limit does not exists.
Some standard result: →
  1. lim x→0 sinx/x = 1, lim x→0 tanx/x = 1
  2. lim x→0 (1-cosx)/x = 0, lim x→0 ex-1/x = 1
  3. lim x→0 ax-1/x = loga.
  4. lim x→0 log(1+x)/x = 1
  5. lim x→a xn-an/x-a = na(n-1)
❈ Differentible function: →
   let f(x) is any function which is defined at x=a then
   LHD = lim h→0 f(a-h)-f(a)/-h
   and RHD = lim h→0 f(a+h)-f(a)/h
Note
   If LHD = RHD ⇒ then f(x) is differentiable
   If LHD ≠ RHD ⇒ then f(x) is not differentiable
✤ SOME IMP QUESTION
  1. Differentiate, sin²x w.r.t. x
  2. If sin y + x = logx find dy/dx
  3. find dy/dx at x=1, y=π/4 If sin²y + cosxy = k
  4. find dy/dx If y = sin⁻¹(√x√1-x + √x√1-x)
  5. Differentiate tan⁻¹(acosx - bsinx) / (bcosx + asinx)
  6. If y = tan⁻¹(√(1-x) - √a) / (1-√(ax)) find dy/dx
  7. If y = sin⁻¹(6x√(1-9x²)), find dy/dx
  8. y = tan⁻¹(√(1+sinx) + √(1-sinx)) /
      √(1+sinx) - √(1-sinx) find dy/dx
  9. If y = tan⁻¹(5au / (a² - 6n²)) prove that
    dy/dx = 3a / (a²+9x²) + 2a / (a²+4x²)
  10. If f(x) = logₑ²(logₑ(x)), then f'(e) = ?
  11. If ny = eˣ⁻ʸ show that dy/dx = y(n-1) / x(y+1)
  12. y = log [e³ˣ(x-3) / (x+3)] find dy/dx
  13. If y = cos⁻¹((2ˣ⁺¹ + 1) / (1+2ˣ)) find dy/dx
  14. y = sin⁻¹((2ˣ⁺¹ + 3ˣ) / (1+26ˣ)) find dy/dx
  15. If y = xˣ⁻¹⁰⁰ then x(1-ylogx)dy/dx = ?
  16. If (cosx) = (sinx) find dy/dx
  17. Differentiate (sinx)ˣ + sin√x w.r.t x
  18. y = (1+1/x)ˣ + x^(x+1/x) w.r.t. x.
  19. diff. sec⁻¹((1/2n²)-1) w.r.t √(1-x²)
  20. y = log (((cosu)ˣ + eˣ²) find dy/dx

FEW IMPORTANT QUESTION

  1. find k if f(x) is continuous at n=0 f(x) = sin(2x) / 5x , n≠0
       k , n = 0
  2. f(x) = (k - xcos x) / (π-2x), n≠π/2 3, n=π/2
       Continuous at x = π/2
  3. find a & b if f(x) = 1, x≤3 ax+b, 3
      is continuous at x=3,5
  4. f(x) = (sin(a+1)x + sinx) / x , x<0 c , n=0
   √(n+bx²)-√n / bx² , n>0
      find a,b,c if it is continuous at n=0
  5. f(x) = (x³-a³) / (n-a), x≠a b, n=a
      Find b if it is continuous at n=a
  6. f(x) = (ant-b, 0≤x<1 2, n=1 x+1, 1
       Find a-b if it is continuous at n=1
  7. Find k if f(x) = (sin 3x) / x , x≠0 k , n=0
       It is continuous at x=0
  8. Find k if f(x) = (x²+3x+10) / (x-2), x≠2 k, n=2
       is continuous at x=2
  9. If y = x^(x^x^...∞) find dy/dx
  10. If y = √sinx + √sinx + √sinx +...∞ find dy/dx
  11. Differentiate lg sinx w.r.t. √cosx
  12. Differentiate sin⁻¹((2x) / (1+x²)) w.r.t. tan⁻¹x
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