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These are two really useful rules for differentiating functions. We use the chain rule when differentiating a 'function of a function', like f(g(x)) in general. We use the product rule when differentiatin...

For the 1st part of the question: use the double angle formula to rewrite cos(2x) = cos^2(x) - sin^2(x). Then use the basic identity to write cos^2(x) = 1-sin^2(x), hence cos(2x) = 1-2sin^2(x). Plug the r...

Split the equation into two : one that you are going to integrate, and the other one you want to differentiate. then, apply formula uv - integral(v * du)

Finding normals and tangents to curves is a very common question in A-level maths papers, especially core 3 modules, giving between 5-8 marks depending on complexity. In order to start this question, the ...

This problem requires using the quotient rule, product rule and the chain rule. The derivative of the entire thing is ((du/dx)v-(dv/dx)u)/v^2 where u=2x+xe^6x and v=9x-2x^2-lnx. dv/dx is relatively strait...