Differentiation and Integration

• If the function is positive, complete the square
• If the function is negative, factorise
• Differentiation multiplies the coefficient by the power and then removes one from the power
• The second derivative is the same process as differentiation but with the already differentiated gradient
• Convex vs Concave lines

  A) Concave

  B) Convex

  
• Local Maximum: +ve, 0, -ve gradient
• Local Minimum: -ve, 0, +ve gradient
• Point of inflection: +ve, 0, +ve gradient
• If the second derivative is positive, it is a minimum point
• If the second derivative is negative, it is a maximum point
• A maximum or minimum point with cut the x axis
• The inflection point will touch the x axis
• A positive gradient will be above the x axis
• A negative gradient will be below the x axis
• A horizontal asymptote be will at the x axis
• Integration is the opposite of differientiation
• In integration, a +c must be added in order to represent any part of the equation we cannot find
• A logarithm is the inverse function to an expotential
• A log with no base is base 10 by default
• ln = loge(x)
• log(x) + log (y) = log (xy)
• log (x) - log (y) = log (x/y)
• loga(x^k) = k*loga(x)
• loga(b + c) DOES NOT EQUAL loga(b) + loga(c)
• (log2x)^3 DOES NOT EQUAL 3log2x
• ln(e) = 1
• e^lnx = x
• Polynomial to Linear

  1. y = ax^n
  2. logy = log (ax^n)
  3. logy = loga + nlogx
  4. y = mx + c
  5. logy = nlogx + loga
  6. n = m, loga = c
• Polynomial to Linear

  1. y = ax^n
  2. logy = log (ax^n)
  3. logy = loga + nlogx
  4. y = mx + c
  5. logy = nlogx + loga
  6. n = m, loga = c
• Exponential to Linear

  1. y = ab^x
  2. logy = logbx + loga
  3. y = mx + c
  4. logb = m, loga = c