Find the derivative of cos x from first principle

Solution:

Let f (x) = cos x.

Accordingly, from the first principle,

f' (x) = limₕ→₀ [f (x + h) - f (x)]/h

f' (x) = limₕ→₀  [cos(x + h) - cos(x)]/h

= limₕ→₀ (cos x cos h - sin x sin h - cos x)/h

= limₕ→₀ [- cos x (1 - cos h) - sin x sin h]/h

= limₕ→₀  [- cos x (1 - cos h)/h - sin x sin h/h]

= - cos x [limₕ→₀ (1 - cos h)/h] - sin x [limₕ→₀ (sin h/h)]

= - cos x (0) - sin x (1) [∵ limₕ→₀ (1 - cos h)/h = 0 and limₕ→₀ (sin h/h) = 1]

= - sin x

NCERT Solutions Class 11 Maths Chapter 13 Exercise 13.2 Question 10

Find the derivative of cos x from first principle

Summary:

The derivative of cos x is -sinx