Find the derivative of the following functions from first principle:

Question 42 of 70intermediate🔧 ApplyLong Answer4 marks

(i)

$-x$

Answer

-1

(ii)

$(-x)^{-1}$

Answer

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

(iii)

$\sin(x + 1)$

Answer

$\cos$ (x + 1)

(iv)

$\cos\left(x - \frac{\pi}{8}\right)$

Answer

- $\sin \left(x - \frac{\pi}{8}\right)$

This question is from Miscellaneous Exercise on Chapter 12, Question 1. The first principle of derivatives uses the definition:

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Each part requires applying this limit definition directly and simplifying using algebraic manipulation and standard trigonometric identities.