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At first this equation seems tricky, but we can perform a clever substitution to simplify it. We notice that if let y = 2^x, then we can rewrite this as:

8(y^2) - 9y + 1 = 0

This now becomes...

A stationary point is where the gradient is exactly zero - the curve is neither increasing or decreasing. This means that we need to differentiate y to find dy/dx and then set this equal to 0. Doing this,...

First write tanx as sinx/cosx as it is always helpful to use what additional information the question gives you. It says we must use the quotient rule to calculate the result so it is also a good idea to ...

Integrate by parts, identifying that dv/dx = 1, u = ln(x) and therefore v = x and du/dx = 1/x. Sub into the parts equation to find the answer to be xln(x) - x + c.

The sine rule is a very powerful tool used in mathematics to calculate unknown sides and angles of triangles when we only know some of the information.

For example if we have a triangle and we know...