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The key concepts to apply in this question will be the product and chain rules, namely: if  f(x)=g(x)h(h), then f'(x)=g(x)h'(x) + g'(x)h(x), and if h(x)=u(v(x)), then h'(x)=u'(v(x))v'(x).

Equivalen...

We start by expanding out the double trigonometric terms (sin(2a)) using the double angle formula, giving us: [2sin(a)cos(a)] / [1+cos^2(a) - sin^2(a)]. Next we spot that on the denominator (bottom half o...

The definite integral gives the value of the area bounded between the curve at the given limits, and the x-axis (if we are integrating with respect to x!) If this value is negative, it means that the area...

A ratio of 1:4 means there are 5 parts to the mixture.
2 litres of drink divided into 5 parts means there are 400 ml per part (2000ml/5 parts).
As Fruit Juice has 1 part, Callum would need 400...

using the product rule

product rule is dy/dx=v.du/dx+u.dv/dx

let u = x^2 and v = cos(x)

du/dx=2x

dv/dx = -sin(x)

so dy/dx=2xcos(x)-x^2sin(x)