If \(u=2x+1\) and \(y=u^3\) what is \(y\) as a function \(x\)?
Working backwards, if \(y=(2x+3)^{-2}\), how could you express \(y\) as a function of \(u\) where \(u\) is a function of \(x\)?
If you also wanted to work out \(\dfrac{dy}{dx}\), how would this affect your choice of \(u\)?
Take a look at the functions in the table below.
Write \(y\) as a function of \(u\) where \(u\) is a function of \(x\) so that you can use these to find \(\dfrac{dy}{dx}.\)
You may be able to do this in more than one way.
| \(y=(5-2x)^3\) | \(y=\sqrt{3x-1}\) | \(y=\dfrac{5}{\sqrt{x}}\) |
| \(y=9x^2-6x+1\) | \(y=e^{5x}\) | \(y=\ln{3x}\) |
| \(y=\dfrac{1}{x^2+4x+4}\) | \(y=e^{x+4}\) | \(y=\ln{x^2}\) |
| \(y=\cot x\) | \(y=x\) | \(y=\ln{x^2}+\ln{8x}\) |
| \(y=\sin x^2\) | \(y=\tan x (\sec^2 x-1)\) | \(y=\tfrac{1}{2}(1-\cos 2x)\) |
