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

Find the derivative of each function.

1. (a) \(f(x)=x^{5/3}\)
2. (b) \(f(x)=12x^{-3/4}\)

Differentiate by rewriting as a power of x.

1. (a) \(f(x)=6\sqrt[3]{x}\)
2. (b) \(f(x)=-\dfrac{9}{\sqrt[4]{x}}\)

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

1. (a) \(y=\dfrac{6x^2+5}{x^{2/3}}\)
2. (b) \(y=\dfrac{7-2x}{\sqrt[5]{x}}\)

Differentiate. The angle is in radians.

1. (a) \(f(x)=5\sin x - 3\cos x\)
2. (b) \(f(x)=8 - \frac{1}{3}\cos x + 6\sin x\)

Differentiate.

1. (a) \(f(x)=4\tan x - 7\)
2. (b) \(f(x)=x^{2} + \tan x\)

Find \(y'\).

1. (a) \(y=6\ln x\)
2. (b) \(y=-5\ln x + 3e^x\)

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

1. (a) \(y=2\ln x^{7}\)
2. (b) \(y=-3\ln x^{-2}\)

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 \(\frac{dh}{dt}\).
2. (b) Find the rate of growth when t = 16.

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 \(\frac{dd}{dt}\).
2. (b) Find the rate of change of depth when t = 3.

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 \(\frac{dy}{d\theta}\).
2. (b) Find the rate of change of y when \(\theta\) = 0.9.

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 \(\frac{dV}{dt}\).
2. (b) Find the rate of change when t = 4.

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

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

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

1. (a) Find \(\dfrac{dy}{dx}\).
2. (b) Find the x-coordinate of the point where the gradient is zero. Give the answer to 3 significant figures.