d)
[tex]y = (2ax^2 + c)^2 (bx^2 - cx)^{-1}[/tex]
Product rule:
[tex]y' = \bigg((2ax^2+c)^2\bigg)' (bx^2-cx)^{-1} + (2ax^2+c)^2 \bigg((bx^2-cx)^{-1}\bigg)'[/tex]
Chain and power rules:
[tex]y' = 2(2ax^2+c)\bigg(2ax^2+c\bigg)' (bx^2-cx)^{-1} - (2ax^2+c)^2 (bx^2-cx)^{-2} \bigg(bx^2-cx\bigg)'[/tex]
Power rule:
[tex]y' = 2(2ax^2+c)(4ax) (bx^2-cx)^{-1} - (2ax^2+c)^2 (bx^2-cx)^{-2} (2bx - c)[/tex]
Now simplify.
[tex]y' = \dfrac{8ax (2ax^2+c)}{bx^2 - cx} - \dfrac{(2ax^2+c)^2 (2bx-c)}{(bx^2-cx)^2}[/tex]
[tex]y' = \dfrac{8ax (2ax^2+c) (bx^2 - cx) - (2ax^2+c)^2 (2bx-c)}{(bx^2-cx)^2}[/tex]
e)
[tex]y = \dfrac{3bx + ac}{\sqrt{ax}}[/tex]
Quotient rule:
[tex]y' = \dfrac{\bigg(3bx+ac\bigg)' \sqrt{ax} - (3bx+ac) \bigg(\sqrt{ax}\bigg)'}{\left(\sqrt{ax}\right)^2}[/tex]
[tex]y'= \dfrac{\bigg(3bx+ac\bigg)' \sqrt{ax} - (3bx+ac) \bigg(\sqrt{ax}\bigg)'}{ax}[/tex]
Power rule:
[tex]y' = \dfrac{3b \sqrt{ax} - (3bx+ac) \left(-\frac12 \sqrt a \, x^{-1/2}\right)}{ax}[/tex]
Now simplify.
[tex]y' = \dfrac{3b \sqrt a \, x^{1/2} + \frac{\sqrt a}2 (3bx+ac) x^{-1/2}}{ax}[/tex]
[tex]y' = \dfrac{6bx + 3bx+ac}{2\sqrt a\, x^{3/2}}[/tex]
[tex]y' = \dfrac{9bx+ac}{2\sqrt a\, x^{3/2}}[/tex]
f)
[tex]y = \sin^2(ax+b)[/tex]
Chain rule:
[tex]y' = 2 \sin(ax+b) \bigg(\sin(ax+b)\bigg)'[/tex]
[tex]y' = 2 \sin(ax+b) \cos(ax+b) \bigg(ax+b\bigg)'[/tex]
[tex]y' = 2a \sin(ax+b) \cos(ax+b)[/tex]
We can further simplify this to
[tex]y' = a \sin(2(ax+b))[/tex]
using the double angle identity for sine.
