Derivatives with product and quotient

Text

Differentiate ($a,b,c,d$ constants):

• (a) $y=(x+a)^{2}$
• (b) $y=(x+a)^{3}$
• (c) $y=\dfrac{2x+3}{3x+2}$
• (d) $y=\dfrac{ax+b}{cx+d}$

Solution

Use the product/power rule and the quotient rule $\left(\tfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^{2}}$.

• (a) $y'=2(x+a)$
• (b) $y'=3(x+a)^{2}$
• (c) $y'=\dfrac{2(3x+2)-3(2x+3)}{(3x+2)^{2}}=\dfrac{-5}{(3x+2)^{2}}$
• (d) $y'=\dfrac{a(cx+d)-c(ax+b)}{(cx+d)^{2}}=\dfrac{ad-bc}{(cx+d)^{2}}$ $\boxed{2(x+a);\ 3(x+a)^{2};\ \tfrac{-5}{(3x+2)^{2}};\ \tfrac{ad-bc}{(cx+d)^{2}}}$