Differentiation of trigonometric functions by first principle Exercise 2.5 Question 1. Class 12 Math

The First Principles Formula
The derivative of a function f(x) from first principles is given by the limit formula: f'(x) = lim h→0 [f(x+h) - f(x)] / h.
1-Differentiating sin(x) from First Principles
2-Identify f(x): Let f(x) = sin(x).
3-Find f(x+h): This gives f(x+h) = sin(x+h).
4-Expand sin(x+h):
Use the trigonometric sum identity sin(A+B) = sin A cos B + cos A sin B to expand sin(x+h) into sin(x)cos(h) + cos(x)sin(h).
5-Substitute into the formula:
f'(x) = lim h→0 [ (sin x cos h + cos x sin h) - sin x ] / h.
Rearrange and factor: Group terms with sin(x) and factor it out:
f'(x) = lim h→0 [ sin x (cos h - 1) + cos x sin h ] / h
Separate into two fractions:
f'(x) = lim h→0 [ sin x (cos h - 1) / h ] + lim h→0 [ cos x sin h / h ] f'(x) = sin x [ lim h→0 (cos h - 1) / h ] + cos x [ lim h→0 sin h / h ].
Apply known limits:
lim h→0 sin h / h = 1
lim h→0 (cos h - 1) / h = 0
Simplify:
f'(x) = sin x (0) + cos x (1)
f'(x) = 0 + cos x f'(x) = cos x.
Therefore, the derivative of sin(x) from first principles is cos(x).