Question 34 of 70intermediate🔧 ApplyLong Answer4 marks

Find the derivative of the following functions from first principle.

(i)

x3−27x^3 - 27

Answer

3x²

(ii)

(x−1)(x−2)(x - 1)(x - 2)

Answer

2x - 3

(iii)

1x2\frac{1}{x^2}

Answer

-2/x³

(iv)

x+1x−1\frac{x + 1}{x - 1}

Answer

-2/(x-1)²

Explanation

This question is from Exercise 12.2, Question 4. Each part requires applying the first principle formula:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The context shows similar examples like Example 19 and Example 20 which demonstrate this method for various functions including rational functions.

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