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

Differentiate each function.

1. (a) \(f(x)=x^8\)

   • \(f'(x)=8x^{7}\)

2. (b) \(h(u)=u^{-5}\)

   • \(h'(u)=-5u^{-5-1}=-5u^{-6}\)

3. (c) \(s=r^{-9}\)

   • \(\frac{ds}{dr}=-9r^{-10}\)

Differentiate each function.

1. (a) \(y=5x^4\)

   • \(5\times4x^3=20x^3\)

2. (b) \(g(t)=-3t^6\)

   • \(-3\times6t^5=-18t^5\)

3. (c) \(y=9\)

   • The derivative of a constant is 0

4. (d) \(y=-4x+7\)

   • The derivative of −4x is −4, and the derivative of 7 is 0

Rewrite as a power of the variable, then differentiate.

1. (a) \(y=\dfrac{5}{x^3}\)

   • \(y=5x^{-3}\)
   • \(\frac{dy}{dx}=5\times(-3)x^{-4}=-15x^{-4}\)

2. (b) \(y=-\dfrac{12}{t^5}\)

   • \(y=-12t^{-5}\)
   • \(\frac{dy}{dt}=-12\times(-5)t^{-6}=60t^{-6}\)

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

   • \(f(x)=-\tfrac23x^{-1}\)
   • \(f'(x)=-\tfrac23\times(-1)x^{-2}=\tfrac23x^{-2}\)

4. (d) \(y=\dfrac{1}{4x^6}\)

   • \(y=\tfrac14x^{-6}\)
   • \(\frac{dy}{dx}=\tfrac14\times(-6)x^{-7}=-\tfrac32x^{-7}\)

Differentiate each function.

1. (a) \(y=4x^3-6x^2+x-9\)

   • Differentiate term by term: 12x², −12x, 1 and 0

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

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

3. (c) \(f(x)=2-3x^2+\tfrac16x^3\)

   • The constant 2 gives 0
   • \(\tfrac16\times3=\tfrac12\)

4. (d) \(y=x^2-3x+\dfrac4x\)

   • \(y=x^2-3x+4x^{-1}\)
   • \(\frac{dy}{dx}=2x-3-4x^{-2}\)

Expand the brackets, then differentiate.

1. (a) \(f(x)=x^2(4x-3)\)

   • \(f(x)=4x^3-3x^2\)
   • \(f'(x)=12x^2-6x\)

2. (b) \(g(x)=(x-5)(x+2)\)

   • \(g(x)=x^2-3x-10\)
   • \(g'(x)=2x-3\)

3. (c) \(h(x)=(3x-1)(x+4)\)

   • \(h(x)=3x^2+11x-4\)
   • \(h'(x)=6x+11\)

Expand the brackets, then differentiate.

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

   • \(h(x)=4+4x^{-1}+x^{-2}\)
   • \(h'(x)=-4x^{-2}-2x^{-3}\)

2. (b) \(g(x)=\left(3x-\dfrac4x\right)^2\)

   • \(g(x)=9x^2-24+16x^{-2}\)
   • The middle term is 2 × 3x × (−4/x) = −24
   • \(g'(x)=18x-32x^{-3}\)

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.

• \(\frac{dh}{dt}=10t^4\)
• \(t=2:\ 10\times 2^4=160\)
• The rate is 160 metres per second

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.

• \(I=90d^{-2}\)
• \(\frac{dI}{dd}=-180d^{-3}\)
• \(d=3:\ -\frac{180}{27}=-6.67\)
• The intensity falls by 6.67 units per metre

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

1. (a) Find \(\frac{dP}{dx}\).

   • \(-\tfrac14\times2=-\tfrac12\)
   • \(\frac{dP}{dx}=-\tfrac12x+30\)

2. (b) Find the rate of change of profit when \(x=40\).

   • \(-\tfrac12\times40+30=10\)

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

   • \(-\tfrac12x+30=0\)
   • \(x=60\)

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.

• \(f'(x)=3ax^2+b\)
• \(f'(1)=3a+b=1\)
• \(f'(2)=12a+b=19\)
• Subtract: 9a = 18, so a = 2, then b = 1 − 6 = −5

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

• \(g'(x)=knx^{n-1}\)
• \(n-1=-3\Rightarrow n=-2\)
• \(kn=-6\Rightarrow -2k=-6\Rightarrow k=3\)