Derivatives with product and quotient

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Compute the derivative of each function:

• (a) $y=(x^{2}+2)(x+3)$
• (b) $y=\dfrac{x^{2}+3}{2x-1}$
• (c) $y=\dfrac{x}{x^{2}+1}$

Solution

(a) Product rule: $y'=2x(x+3)+(x^{2}+2)=3x^{2}+6x+2$.

(b) Quotient rule: $y'=\dfrac{2x(2x-1)-(x^{2}+3)\cdot 2}{(2x-1)^{2}}=\dfrac{2x^{2}-2x-6}{(2x-1)^{2}}$.

(c) $y'=\dfrac{(x^{2}+1)-x\cdot 2x}{(x^{2}+1)^{2}}=\dfrac{1-x^{2}}{(x^{2}+1)^{2}}$.