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Grouping is essentially factorisation, it's just the idea that ax + bx = (a+b)x, which we already know. But the difficulty is in recognising this relationship when x is more complicated, for example when ...

Multiply out the brackets:

3a+12=ac+5f

3a-ac=5f-12

a(3-c)=5f-12

a=5f-12/3-c

The first step is to split the fraction into 2 separate fractions using partial fractions techniques. Write 3x+3/2x^2+3x as A/x + B/2x+3 and solve to get A = 1, B = 1. We have now converted 3x+3/2x^2+3x i...

In order to find the points where these functions meet, we can equate them to get 3x-2 = x^2+4x-8 .

Subtraction (3x-2) from both sides, we get x^2+x-6 =0 which we can factorise to get (x+3)(x-2)=0<...

We have a "fraction" which we wish to differentiate, so we use the quotient rule with u=sin(x) and v=cos(x).

This means that d/dx of u/v = (vdu/dx - udv/dx)/(v^2).