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

Find the derivative of each function.

1. (a) \(f(x)=x^{7/4}\)
2. (b) \(f(x)=15x^{-2/5}\)

Differentiate by rewriting as a power of x.

1. (a) \(f(x)=8\sqrt[5]{x}\)
2. (b) \(f(x)=-\dfrac{14}{\sqrt[3]{x}}\)

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

1. (a) \(y=\dfrac{10x^3+3}{x^{3/2}}\)
2. (b) \(y=\dfrac{9+4x}{\sqrt[3]{x}}\)

Differentiate. The angle is in radians.

1. (a) \(f(x)=4\sin x + 7\cos x\)
2. (b) \(f(x)=3 + \frac{1}{4}\sin x - 5\cos x\)

Differentiate.

1. (a) \(f(x)=-6\tan x + 2\)
2. (b) \(f(x)=x^{3} + \tan x\)

Find \(y'\).

1. (a) \(y=8\ln x\)
2. (b) \(y=-3\ln x + 5e^x\)

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

1. (a) \(y=5\ln x^{3}\)
2. (b) \(y=-2\ln x^{-4}\)

The volume of water in a tank filling slowly is modelled by \(V=12t^{3/2}\) litres, where t is the time in minutes since the tap was opened (t > 0).

1. (a) Find \(\frac{dV}{dt}\).
2. (b) Find the rate of change of volume when t = 9.

The temperature of an oven is modelled by \(T=180+12\sin t-9\cos t\) degrees Celsius, where t is the time in minutes after it was switched on. The angle is in radians.

1. (a) Find \(\frac{dT}{dt}\).
2. (b) Find the rate of change of temperature when t = 1.5.

A laser pointer on a stand is 40 cm from a wall and turns through an angle of \(\theta\) radians. The dot on the wall is at a distance \(x=40\tan\theta\) centimetres from the point nearest the stand.

1. (a) Find \(\frac{dx}{d\theta}\).
2. (b) Find the rate of change of x when \(\theta\) = 0.7.

The size of a fish population, in hundreds, is modelled by \(N=30+8\ln w^2\), where w is the number of weeks since a pond was stocked (w > 0).

1. (a) Use a law of logarithms to write \(N\) without a power inside the logarithm, then find \(\frac{dN}{dw}\).
2. (b) Find the rate of change when w = 5.

The curve \(y=f(x)\) has equation \(f(x)=\dfrac{x^2-12x}{\sqrt{x}}\), 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 \(y=e^x-6\ln x\) 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.