Chain, product and quotient rules

Mark scheme

1. Answer. $15(3x - 1)^4$

   • Use the chain rule: $5(3x - 1)^4 \times 3$.

2. Answer. $e^{2x}(1 + 2x)$

   • Use the product rule: $e^{2x} + x \times 2e^{2x}$.

3. Answer. $\dfrac{x\cos x - \sin x}{x^2}$

   • Use the quotient rule with $u = \sin x$ and $v = x$.

4. Answer. $\dfrac{6x}{3x^2 + 1}$

   • The derivative of $\ln u$ is $\dfrac{1}{u} \times u'$.
   • $u' = 6x$.

5. Answer. $-\sin x\, e^{\cos x}$

   • Use the chain rule.
   • The derivative of the inner function $\cos x$ is $-\sin x$.

6. Answer. $\dfrac{e^{2x}(2x + 1)}{(x + 1)^2}$

   • The quotient rule gives $\dfrac{2e^{2x}(x + 1) - e^{2x}}{(x + 1)^2}$.
   • Factorise the numerator as $e^{2x}(2x + 1)$.

7. Answer. $6\sin^2 2x \cos 2x$

   • Write the function as $(\sin 2x)^3$.
   • The chain rule gives $3\sin^2 2x \times \cos 2x \times 2$.

8. Answer. $\dfrac{x(3x + 4)}{2(x + 1)^{3/2}}$

   • Write the function as $x^2(x + 1)^{-1/2}$.
   • The product rule gives $2x(x + 1)^{-1/2} - \dfrac{1}{2}x^2(x + 1)^{-3/2}$.
   • Factor $x(x + 1)^{-3/2}$ to obtain $\dfrac{x(3x + 4)}{2(x + 1)^{3/2}}$.