Differentiating powers of x Problem set B One question at a time

Differentiate each function.

1. (a) \(g(x)=x^7\)

   • \(g'(x)=7x^{6}\)

2. (b) \(z(t)=t^{-8}\)

   • \(z'(t)=-8t^{-8-1}=-8t^{-9}\)

3. (c) \(p=q^{-7}\)

   • \(\frac{dp}{dq}=-7q^{-8}\)

Differentiate each function.

1. (a) \(y=6x^5\)

   • \(6\times5x^4=30x^4\)

2. (b) \(g(u)=-2u^7\)

   • \(-2\times7u^6=-14u^6\)

3. (c) \(y=-8\)

   • The derivative of a constant is 0

4. (d) \(y=3x-10\)

   • The derivative of 3x is 3, and the derivative of −10 is 0

Rewrite as a power of the variable, then differentiate.

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

   • \(y=7x^{-2}\)
   • \(\frac{dy}{dx}=7\times(-2)x^{-3}=-14x^{-3}\)

2. (b) \(y=-\dfrac{9}{t^4}\)

   • \(y=-9t^{-4}\)
   • \(\frac{dy}{dt}=-9\times(-4)t^{-5}=36t^{-5}\)

3. (c) \(f(x)=-\dfrac{5}{2x}\)

   • \(f(x)=-\tfrac52x^{-1}\)
   • \(f'(x)=-\tfrac52\times(-1)x^{-2}=\tfrac52x^{-2}\)

4. (d) \(y=\dfrac{1}{5x^5}\)

   • \(y=\tfrac15x^{-5}\)
   • \(\frac{dy}{dx}=\tfrac15\times(-5)x^{-6}=-x^{-6}\)

Differentiate each function.

1. (a) \(y=5x^3-8x^2+x-4\)

   • Differentiate term by term: 15x², −16x, 1 and 0

2. (b) \(y=\tfrac34x^4-\tfrac25x^5\)

   • \(\tfrac34\times4=3\)
   • \(\tfrac25\times5=2\)

3. (c) \(f(x)=7-2x^2+\tfrac13x^3\)

   • The constant 7 gives 0
   • \(\tfrac13\times3=1\)

4. (d) \(y=x^3-2x+\dfrac6x\)

   • \(y=x^3-2x+6x^{-1}\)
   • \(\frac{dy}{dx}=3x^2-2-6x^{-2}\)

Expand the brackets, then differentiate.

1. (a) \(f(x)=x^3(5x-2)\)

   • \(f(x)=5x^4-2x^3\)
   • \(f'(x)=20x^3-6x^2\)

2. (b) \(g(x)=(x-6)(x+4)\)

   • \(g(x)=x^2-2x-24\)
   • \(g'(x)=2x-2\)

3. (c) \(h(x)=(2x-3)(x+5)\)

   • \(h(x)=2x^2+7x-15\)
   • \(h'(x)=4x+7\)

Expand the brackets, then differentiate.

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

   • \(h(x)=9+6x^{-1}+x^{-2}\)
   • \(h'(x)=-6x^{-2}-2x^{-3}\)

2. (b) \(g(x)=\left(2x-\dfrac5x\right)^2\)

   • \(g(x)=4x^2-20+25x^{-2}\)
   • The middle term is 2 × 2x × (−5/x) = −20
   • \(g'(x)=8x-50x^{-3}\)

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.

• \(\frac{dN}{dt}=12t^3\)
• \(t=3:\ 12\times 27=324\)
• The rate is 324 thousand cells per hour

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.

• \(F=360r^{-3}\)
• \(\frac{dF}{dr}=-1080r^{-4}\)
• \(r=3:\ -\frac{1080}{81}=-13.3\)
• The force falls by 13.3 newtons per thousand km

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

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

   • \(-\tfrac34\times2=-\tfrac32\)
   • \(\frac{dh}{dt}=-\tfrac32t+18\)

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

   • \(-\tfrac32\times4+18=12\)

3. (c) Find the value of t for which the rate of change is 0.

   • \(-\tfrac32t+18=0\)
   • \(t=12\)

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.

• \(f'(x)=3px^2+q\)
• \(f'(1)=3p+q=5\)
• \(f'(2)=12p+q=41\)
• Subtract: 9p = 36, so p = 4, then q = 5 − 12 = −7

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

• \(g'(x)=knx^{n-1}\)
• \(n-1=-5\Rightarrow n=-4\)
• \(kn=-20\Rightarrow -4k=-20\Rightarrow k=5\)