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Using the chain rule of dy/dx=dy/du * du/dx we label sin(x) as u. Now we differentiate u with respect to x, getting cos(x). Then we differentiate u 2 , getting 2u. Mutiplying these together gets us 2u*cos(x)...

We start with: 2ln6 - ln3 ... First, we rewrite this expression as: ln6 + ln6 - ln3 ... Next, we rewrite this as: ln(2 3) + ln(2 3) - ln3 ... Using the log rule log a xy = log a x + log x y, we express this ...

The first step in scoring full marks on this typically 4 mark question is to recognise what it's asking you to do. We use the process of differentiation to solve it. f(x)=2x^3 - 2x^2 + 8xf'(x) = 6x^2 - 4x + ...

We need to split the initial fraction into two. We know that (x-5)/(x+1)(x-2) is equivalent to A/(x+1) + B/(x-2). Multiplying by (x+1)(x-2) we get x-5 = A(x-2) + B(x+1). Expanding the brackets gives x - 5 = ...

This is a classic "picture frame" resolving forces question. By resolving horizontally we can use the "assuming the particle remains stationary" meaning the forces must cancel out, so, xC...