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Pure Maths (Logarithms (Rules (When you have two logs that are added…

- - - - f(x) = e^kx, then f'(x) = ke^kx
    - - e^x is good to model situations with rates of growth, ie population growth. Similarly e^-x can model exponential decay, ie radioactive decay.
  - - - Similarly, when you are subtracting two logarithms, use the division law.
- - - - Finally, the first derivative can help us find any stationary points. Set f'(x) = 0 and solve to find x. Then substitute the x value back into your original equation to find the y -coordinate. (if you substitute it back into the derivative, you'll just end up with 0).
        
        Remember that any stationary/turning point has a gradient of zero, so before you start finding any turning points it is important to set the first derivative / f'(x) to 0.
        
        If you differentiate the equation again for the second time (known as the second derivative) you will be able to find the rate of change of the gradient function, or quite simply, the nature of the turning point.
        
        If f''(x) > 0, the point is a local minimum.
        
        If f''(x) < 0, the point is a local maximum.
        
        Why? f''(x) > 0 is a minimum? Yes that's because the curve is increasing from a minimum point.
        
        if f''(a) = 0, the point is either a point of minimum, maximum or even a point of inflection (where the graph touches the x-axis but does immediately not cut through it).
      - As well as finding the gradient of a given point along the curve, the differentiated equation can be used to determined weather a function is increasing or decreasing.
        
        If the derivative is...
        ≥ 0, the graph is increasing for all real values of x
        ≤ 0, the graph is decreasing for all real values of x
  - - - Area under graphs? Yes! This is very useful, since the area under a velocity time graph is distance, so integration will help us find the distance travelled.
      - The first part of integration is simple :cat2:, because its the revers of differentiation, you add one to the power, and divide the corresponding constant by this new power. The write +c at the end of the equation because there is a missing constant.
        
        Edexcel makes find the constant really straigtforwrad for you becuase an exam question will always give you a point on the graph. Simply substitute these values into the first derivative and solve to find c.
      - Hint! When integrating polynomials, apply the rule separatley to each term.