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1 - Derivative of sin x.
The derivative of f(x) = sin x is given by
f '(x) = cos x
2 - Derivative of cos x.
The derivative of f(x) = cos x is given by
f '(x) = - sin x
3 - Derivative of tan x.
The derivative of f(x) = tan x is given by
f '(x) = sec 2 x
4 - Derivative of cot x.
The derivative of f(x) = cot x is given by
f '(x) = - csc 2 x
5 - Derivative of sec x.
The derivative of f(x) = sec x tan x is given by
f '(x) = sec x tan x
6 - Derivative of csc x.
The derivative of f(x) = csc xis given by
f '(x) = - csc x cot x
Example 1: Find the first derivative of f(x) = x sin x
Solution to Example 1:
- Let g(x) = x and h(x) = sin x, function f may be considered as the product of functions g and h: f(x) = g(x) h(x). Hence we use the product rule, f '(x) = g(x) h '(x) + h(x) g '(x), to differentiate function f as follows
f '(x) = x cos x + sin x * 1 = x cos x + sin x
Example 2: Find the first derivative of f(x) = tan x + sec x
Solution to Example 2:
- Let g(x) = tan x and h(x) = sec x, function f may be considered as the sum of functions g and h: f(x) = g(x) + h(x). Hence we use the sum rule, f '(x) = g '(x) + h '(x), to differentiate function f as follows
f '(x) = sec 2 x + sec x tan x = sec x (sec x + tan x)
Example 3: Find the first derivative of f(x) = sin x / [ 1 + cos x ]
Solution to Example 3:
- Let g(x) = sin x and h(x) = 1 + cos x, function f may be considered as the quotient of functions g and h: f(x) = g(x) / h(x). Hence we use the quotient rule, f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x) 2, to differentiate function f as follows
g '(x) = cos x
h '(x) = - sin x
f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x) 2
= [ (1 + cos x)(cos x) - (sin x)(- sin x) ] / (1 + cos x) 2
= [ cos x + cos 2x + sin 2x ] / (1 + cos x) 2
- Use trigonometric identity cos 2x + sin 2x = 1 to simplify the above
f '(x) = [ cos x + 1 ] / (1 + cos x) 2 = 1 / [cos x + 1]
More references on
Differentiation
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