The product and quotient rules Problem set B Answers

1. (a) \(x^6(7\cos x-x\sin x)\)
2. (b) \(e^x\left(x^{5/2}+\tfrac52x^{3/2}\right)\)
3. (c) \(x^7(8\ln x+1)\)

1. (a) \(2x\cos 5x-5x^2\sin 5x\)
2. (b) \(3x^2e^{3x+4}(1+x)\)
3. (c) \(2x\ln(4x+1)+\dfrac{4x^2}{4x+1}\)

1. (a) \(\dfrac{-4x^2+6x+20}{(x^2+5)^2}\)
2. (b) \(\dfrac{x+6}{(2x+1)^{3/2}}\)
3. (c) \(\dfrac{e^x(5x-8)}{(5x+2)^3}\)
4. (d) \(\dfrac{e^x(6-x)}{(2-x)^5}\)

1. (a) 22.2
2. (b) \(3x^2\sin x\ln x+x^3\cos x\ln x+x^2\sin x\)
3. (c) \(e^{x\sin2x}(\sin2x+2x\cos2x)\)

1. (a) \(y=\dfrac{x+1}{8}\)
2. (b) \(y=\dfrac{2x}{\pi}-\dfrac12\)

1. (a) \((0.366,\;1.54)\)
2. (b) \((6.70,\;20.1)\)

1. (a) \(\dfrac{dP}{dt}=8e^{-0.25t}(1-0.25t)\)
2. (b) t = 4 months; P = 11.8 thousand pounds
3. (c) −1.08 thousand pounds per month, so the profit is falling

1. (a) \(\dfrac{dV}{dt}=\dfrac{30(25-t^2)}{(t^2+25)^2}\)
2. (b) 0.415 thousand visitors per day
3. (c) t = 5 days; V = 3 thousand

1. (a) \(a=9,\; b=2\)
2. (b) \(\dfrac{k}{(k+2x^2)^{3/2}}\)

\(a=8,\; b=0\)

\(f'(x)=\dfrac{\sin x}{\cos^2x}\)

1. (a) \(\dfrac{dy}{dx}=4\sin x\cos x\)
2. (b) \(x=\dfrac{\pi}{2},\; \pi,\; \dfrac{3\pi}{2}\)
3. (c) \(\sin2x=2\sin x\cos x\)
4. (d) \(\cos4x=\cos^22x-\sin^22x\)

1. (a) \(\dfrac{dy}{dx}=4e^{4x}=4y\)
2. (b) No. For example \(y=7e^{4x}\) also has \(\dfrac{dy}{dx}=4y\), and it is not \(e^{4x}\)
3. (c) y-intercept \(=-2e^3\), which does not contain k

\(x=2.81\)