AI SL · 12.9
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Differentiating powers of x
12.9.5
Given \(f(x)=x^6\), find \(f'(x)\). Then find \(f'(2)\).
Have a go first. Then open the working one line at a time.
\(f(x)=x^n \Rightarrow f'(x)=nx^{n-1}\)
Bring the power down in front, then take 1 off the power.
\(f'(x)=6x^{5}\)
n = 6, so the new power is 5.
\(f'(2)=6\times 2^5=192\)
2⁵ = 32 and 6 × 32 = 192.
A curve has equation \(y=x^{-3}\). Find \(\frac{dy}{dx}\) when \(x=2\).
Have a go first. Then open the working one line at a time.
\(\frac{dy}{dx}=-3x^{-3-1}=-3x^{-4}\)
n = −3. Taking 1 off −3 gives −4.
\(-3\times 2^{-4}=-\frac{3}{16}=-0.1875\)
2⁻⁴ = 1/16.
Find the derivative of \(f(t)=4t^7\), and evaluate it at \(t=2\).
Have a go first. Then open the working one line at a time.
\(t^7\to 7t^6\)
Power rule on its own.
\(f'(t)=4\times 7t^6=28t^6\)
The constant multiplier stays.
\(f'(2)=28\times 64=1792\)
2⁶ = 64.
Differentiate \(y=\dfrac{8}{x^3}\). Find the gradient when \(x=2\).
Have a go first. Then open the working one line at a time.
\(y=8x^{-3}\)
Use the laws of indices first: 1/xⁿ = x⁻ⁿ.
\(\frac{dy}{dx}=8\times(-3)x^{-4}=-24x^{-4}\)
Multiply 8 by −3, then take 1 off the power.
\(-24\times 2^{-4}=-\frac{24}{16}=-1.5\)
2⁻⁴ = 1/16.
Find \(\frac{dy}{dx}\) for \(y=-6x+11\).
Have a go first. Then open the working one line at a time.
\(y=-6x^1+11x^0\)
x = x¹ and a constant is a multiple of x⁰.
\(\frac{dy}{dx}=-6\times 1x^0+11\times 0x^{-1}=-6\)
The derivative of ax is a. The derivative of a constant is 0. You can use both without the working.
Find \(f'(x)\) for \(f(x)=2x^3-\frac{3}{4}x^4+5x-7\). Then find \(f'(2)\).
Have a go first. Then open the working one line at a time.
\(2x^3\to 6x^2,\ \ -\tfrac34x^4\to -3x^3,\ \ 5x\to 5,\ \ -7\to 0\)
For the fraction, multiply: (3/4) × 4 = 3.
\(f'(x)=6x^2-3x^3+5\)
Differentiate each term on its own.
\(f'(2)=6\times4-3\times8+5=5\)
24 − 24 + 5.
Find \(\frac{dy}{dx}\) for \(y=(2x-3)(x+4)\). Then find its value when \(x=3\).
Have a go first. Then open the working one line at a time.
\(y=2x^2+8x-3x-12=2x^2+5x-12\)
You cannot differentiate a product of brackets term by term. Multiply out first.
\(\frac{dy}{dx}=4x+5\)
Power rule on each term.
\(4\times3+5=17\)
12 + 5.
Find \(f'(x)\) for \(f(x)=\left(x+\dfrac{2}{x}\right)^2\). Then find \(f'(2)\).
Have a go first. Then open the working one line at a time.
\(f(x)=x^2+2\times x\times\frac2x+\frac{4}{x^2}=x^2+4+4x^{-2}\)
The middle term is 2 × x × 2/x = 4. Write 4/x² as 4x⁻².
\(f'(x)=2x-8x^{-3}\)
The constant 4 gives 0. 4 × (−2) = −8.
\(f'(2)=4-8\times\frac18=3\)
2⁻³ = 1/8.
Let \(f(x)=ax^3-4x\), where a is a constant. Given that \(f'(2)=20\), find a.
Have a go first. Then open the working one line at a time.
\(f'(x)=3ax^2-4\)
a is just a number. Treat it like 5 or 7: 3 × a stays 3a.
\(3a\times 2^2-4=20\), so \(12a=24\)
Substitute x = 2, then solve for a.
\(a=2\)
Check: f'(x) = 6x² − 4, and f'(2) = 24 − 4 = 20.