Differentiating powers of x Problem set A

Differentiate each function.

1. (a) \(f(x)=x^8\)
2. (b) \(h(u)=u^{-5}\)
3. (c) \(s=r^{-9}\)

Differentiate each function.

1. (a) \(y=5x^4\)
2. (b) \(g(t)=-3t^6\)
3. (c) \(y=9\)
4. (d) \(y=-4x+7\)

Rewrite as a power of the variable, then differentiate.

1. (a) \(y=\dfrac{5}{x^3}\)
2. (b) \(y=-\dfrac{12}{t^5}\)
3. (c) \(f(x)=-\dfrac{2}{3x}\)
4. (d) \(y=\dfrac{1}{4x^6}\)

Differentiate each function.

1. (a) \(y=4x^3-6x^2+x-9\)
2. (b) \(y=\tfrac25x^5-\tfrac34x^4\)
3. (c) \(f(x)=2-3x^2+\tfrac16x^3\)
4. (d) \(y=x^2-3x+\dfrac4x\)

Expand the brackets, then differentiate.

1. (a) \(f(x)=x^2(4x-3)\)
2. (b) \(g(x)=(x-5)(x+2)\)
3. (c) \(h(x)=(3x-1)(x+4)\)

Expand the brackets, then differentiate.

1. (a) \(h(x)=\left(2+\dfrac1x\right)^2\)
2. (b) \(g(x)=\left(3x-\dfrac4x\right)^2\)

A drone climbs so that its height after t seconds is \(h=2t^5\) metres, for \(t\ge0\). Find the rate of change of height, \(\frac{dh}{dt}\), when \(t=2\). Give the unit.

The intensity of light from a lamp is \(I=\dfrac{90}{d^2}\) units at a distance d metres from it. Find \(\frac{dI}{dd}\) when \(d=3\), to 3 significant figures.

A company's profit is \(P=-\tfrac14x^2+30x-150\) thousand euros when it sells x hundred items.

1. (a) Find \(\frac{dP}{dx}\).
2. (b) Find the rate of change of profit when \(x=40\).
3. (c) Find the value of x for which the rate of change is 0.

The function \(f(x)=ax^3+bx\), where a and b are constants, has \(f'(1)=1\) and \(f'(2)=19\). Find a and b.

The function \(g(x)=kx^n\), where k and n are constants, has derivative \(g'(x)=-6x^{-3}\). Find k and n.