Y2 calculus

• Differentiating rules

  • ln(x) = 1/x
  • sinkx = kcoskx
  • coskx = -ksinkx
  • e^kx = ke^kx
  • a^kx = a^kx * kln(a)
• chain rule

  dy/dx = dy/du * du/dx

  e.g. y = (3x^4 + x)^5

  • bring down the power of 5 --> 5(3x...
  • multiply by inside of brackets differentiated
  • dy/dx = 5(3x^4 + x)^4(12x^3 + 1)
• differentials (in formula booklet)

  • tankx = ksec^2(kx)
  • coseckx = -kcosec(kx)cot(kx)
  • seckx = ksec(kx)tan(kx)
  • cotkx = -kcosec^2(kx)
• integration rules (don't forget +c)

  • x^n = x^(n+1)/n+1
  • e^x = e^x
  • 1/x = ln|x| (modulus)
  • cosx = sin(x)
  • sinx = -cos(x)
  • sec^2(x) = tan(x)
  • cosec(x)cot(x) = -cosec(x)
  • cosec^2(x) = -cot(x)
  • sec(x)tan(x) = sec(x)
• integration by substitution

  • we are given an expression for u in terms of x
  • differentiate (du/dx) to find an expression for dx in terms of du
  • use this expression to rewrite the integral in the du form
  • integrate this expression and substitute in x
• product rule

  for y = uv

  dy/dx = uv' + vu'
• quotient rule

  for y = u/v

  dy/dx = (vu'-uv')/v^2