Differentiating powers, trig, e^x and ln x Problem set A Worked solutions

Find the derivative of each function.

1. (a) \(f(x)=x^{5/3}\)

   • \(f(x)=x^{5/3}\)
   • \(f'(x)=\frac{5}{3}\,x^{5/3-1}\)
   • \(f'(x)=\frac{5}{3}x^{2/3}\)

2. (b) \(f(x)=12x^{-3/4}\)

   • \(f(x)=12x^{-3/4}\)
   • \(f'(x)=12\times (-\frac{3}{4})\,x^{-3/4-1}\)
   • \(f'(x)=-9x^{-7/4}\)

Differentiate by rewriting as a power of x.

1. (a) \(f(x)=6\sqrt[3]{x}\)

   • \(f(x)=6\sqrt[3]{x}=6x^{1/3}\)
   • \(f'(x)=2x^{-2/3}\)

2. (b) \(f(x)=-\dfrac{9}{\sqrt[4]{x}}\)

   • \(f(x)=-\dfrac{9}{\sqrt[4]{x}}=-9x^{-1/4}\)
   • \(f'(x)=\frac{9}{4}x^{-5/4}\)

Find \(\dfrac{dy}{dx}\) by splitting the fraction into powers of x.

1. (a) \(y=\dfrac{6x^2+5}{x^{2/3}}\)

   • \(y=6x^{4/3} + 5x^{-2/3}\quad\text{(split into powers)}\)
   • \(\frac{dy}{dx}=8x^{1/3} - \frac{10}{3}x^{-5/3}\)

2. (b) \(y=\dfrac{7-2x}{\sqrt[5]{x}}\)

   • \(y=7x^{-1/5} - 2x^{4/5}\quad\text{(split into powers)}\)
   • \(\frac{dy}{dx}=-\frac{7}{5}x^{-6/5} - \frac{8}{5}x^{-1/5}\)

Differentiate. The angle is in radians.

1. (a) \(f(x)=5\sin x - 3\cos x\)

   • \(f'(x)=5\cos x + 3\sin x\)

2. (b) \(f(x)=8 - \frac{1}{3}\cos x + 6\sin x\)

   • \(f'(x)=\frac{1}{3}\sin x + 6\cos x\)

Differentiate.

1. (a) \(f(x)=4\tan x - 7\)

   • \(f'(x)=\frac{4}{\cos^2 x}\)

2. (b) \(f(x)=x^{2} + \tan x\)

   • \(f'(x)=2x + \frac{1}{\cos^2 x}\)

Find \(y'\).

1. (a) \(y=6\ln x\)

   • \(y'=\frac{6}{x}\)

2. (b) \(y=-5\ln x + 3e^x\)

   • \(y'=-\frac{5}{x} + 3e^x\)

Use a law of logarithms first, then find \(y'\).

1. (a) \(y=2\ln x^{7}\)

   • \(y=2\ln x^{7}=14\ln x\)
   • \(y'=\frac{14}{x}\)

2. (b) \(y=-3\ln x^{-2}\)

   • \(y=-3\ln x^{-2}=6\ln x\)
   • \(y'=\frac{6}{x}\)

The height of a bamboo shoot is modelled by \(h=20t^{3/4}\) cm, where t is the time in days since it was planted (t > 0).

1. (a) Find \(\frac{dh}{dt}\).

   • \(h=20t^{3/4}\)
   • \(\frac{dh}{dt}=15t^{-1/4}\)

2. (b) Find the rate of growth when t = 16.

   • \(\frac{dh}{dt}\) at t = 16: 7.5 cm per day

The depth of water in a harbour is modelled by \(d=6+2\sin t+1.5\cos t\) metres, where t is the time in hours after midnight. The angle is in radians.

1. (a) Find \(\frac{dd}{dt}\).

   • \(d=6 + 2\sin t + \frac{3}{2}\cos t\)
   • \(\frac{dd}{dt}=2\cos t - \frac{3}{2}\sin t\)

2. (b) Find the rate of change of depth when t = 3.

   • Calculator in radian mode.
   • \(\frac{dd}{dt}\) at t = 3: −2.19 m per hour

A searchlight on a boat shines on a vertical cliff 25 m away. The beam hits the cliff at a height of \(y=25\tan\theta\) metres above the level of the lamp, where \(\theta\) is the angle of the beam in radians.

1. (a) Find \(\frac{dy}{d\theta}\).

   • \(y=25\tan \theta\)
   • \(\frac{dy}{d\theta}=\frac{25}{\cos^2 \theta}\)

2. (b) Find the rate of change of y when \(\theta\) = 0.9.

   • Calculator in radian mode.
   • \(\frac{dy}{d\theta}\) at \(\theta\) = 0.9: 64.7 m per radian

The value of a rare coin, in thousands of dollars, is modelled by \(V=12+5\ln t^3\), where t is the number of years since it was bought (t > 0).

1. (a) Use a law of logarithms to write \(V\) without a power inside the logarithm, then find \(\frac{dV}{dt}\).

   • \(\ln t^{3}=3\ln t\)
   • \(V=12 + 15\ln t\)
   • \(\frac{dV}{dt}=\frac{15}{t}\)

2. (b) Find the rate of change when t = 4.

   • \(\frac{dV}{dt}\) at t = 4: 3.75 thousand dollars per year

The curve \(y=f(x)\) has equation \(f(x)=\dfrac{x^2-6x}{\sqrt{x}}\), for x > 0.

1. (a) Find \(f'(x)\).

   • \(f(x)=x^{3/2} - 6x^{1/2}\)
   • \(f'(x)=\frac{3}{2}x^{1/2} - 3x^{-1/2}\)

2. (b) Find the value of x at the stationary point of the curve, where \(f'(x)=0\).

   • \(f'(x)=0:\ \frac{3}{2}x^{1/2} - 3x^{-1/2}=0\)
   • Multiply by \(x^{3/2}\) or solve with the calculator: x = 2

The curve \(y=e^x-4\ln x\) is defined for x > 0.

1. (a) Find \(\dfrac{dy}{dx}\).

   • \(\frac{dy}{dx}=e^x - \frac{4}{x}\)

2. (b) Find the x-coordinate of the point where the gradient is zero. Give the answer to 3 significant figures.

   • \(e^x-\frac{4}{x}=0\)
   • Solve: x = 1.20 (the only positive solution)