Differentiating powers of x Problem set B

Differentiate each function.

1. (a) \(g(x)=x^7\)
2. (b) \(z(t)=t^{-8}\)
3. (c) \(p=q^{-7}\)

Differentiate each function.

1. (a) \(y=6x^5\)
2. (b) \(g(u)=-2u^7\)
3. (c) \(y=-8\)
4. (d) \(y=3x-10\)

Rewrite as a power of the variable, then differentiate.

1. (a) \(y=\dfrac{7}{x^2}\)
2. (b) \(y=-\dfrac{9}{t^4}\)
3. (c) \(f(x)=-\dfrac{5}{2x}\)
4. (d) \(y=\dfrac{1}{5x^5}\)

Differentiate each function.

1. (a) \(y=5x^3-8x^2+x-4\)
2. (b) \(y=\tfrac34x^4-\tfrac25x^5\)
3. (c) \(f(x)=7-2x^2+\tfrac13x^3\)
4. (d) \(y=x^3-2x+\dfrac6x\)

Expand the brackets, then differentiate.

1. (a) \(f(x)=x^3(5x-2)\)
2. (b) \(g(x)=(x-6)(x+4)\)
3. (c) \(h(x)=(2x-3)(x+5)\)

Expand the brackets, then differentiate.

1. (a) \(h(x)=\left(3+\dfrac1x\right)^2\)
2. (b) \(g(x)=\left(2x-\dfrac5x\right)^2\)

A culture of bacteria has \(N=3t^4\) thousand cells after t hours. Find the rate of change of N, \(\frac{dN}{dt}\), when \(t=3\). Give the unit.

The force of gravity on a probe is \(F=\dfrac{360}{r^3}\) newtons at a distance r thousand km from a planet. Find \(\frac{dF}{dr}\) when \(r=3\), to 3 significant figures.

The height of a model rocket is \(h=-\tfrac34t^2+18t+2\) metres, t seconds after launch.

1. (a) Find \(\frac{dh}{dt}\).
2. (b) Find the rate of change of height when \(t=4\).
3. (c) Find the value of t for which the rate of change is 0.

The function \(f(x)=px^3+qx\), where p and q are constants, has \(f'(1)=5\) and \(f'(2)=41\). Find p and q.

The function \(g(x)=kx^n\), where k and n are constants, has derivative \(g'(x)=-20x^{-5}\). Find k and n.