Formulas of the derivatives, in calculus, of inverse trigonometric functions are presented along with several other examples involving sums, products and quotients of functions.
1 - Derivative of arcsin x.
The derivative of f(x) = arcsin x is given by
f '(x) = 1 / √(1 - x^{ 2})
2 - Derivative of arccos x.
The derivative of f(x) = arccos x is given by
f '(x) = - 1 / √(1 - x^{ 2})
3 - Derivative of arctan x.
The derivative of f(x) = arctan x is given by
f '(x) = 1 / (1 + x^{ 2})
4 - Derivative of arccot x.
The derivative of f(x) = arccot x is given by
f '(x) = - 1 / (1 + x^{ 2})
5 - Derivative of arcsec x.
The derivative of f(x) = arcsec x tan x is given by
f '(x) = 1 / (x √(x^{ 2} - 1))
6 - Derivative of arccsc x.
The derivative of f(x) = arccsc x is given by
f '(x) = - 1 / (x √(x^{ 2} - 1))
Example 1: Find the derivative of f(x) = x arcsin x
Solution to Example 1:
Let h(x) = x and g(x) = arcsin x, function f is considered as the product of functions h and g: f(x) = h(x) g(x). Hence we use the product rule, f '(x) = h(x) g '(x) + g(x) h '(x), to differentiate function f as follows
f '(x) = x (1 / √(1 - x^{ 2})) + arcsin x * 1 = x / √(1 - x^{ 2}) + arcsin x
Example 2: Find the first derivative of f(x) = arctan x + x^{ 2} Solution to Example 2:
Let g(x) = arctan x and h(x) = x^{ 2}, 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) = 1 / (1 + x^{ 2}) + 2x = (2x^{ 3} + 2x + 1) / (1 + x^{ 2})
Example 3: Find the first derivative of f(x) = arcsin (2x + 2)
Solution to Example 3:
Let u(x) = 2x + 2, function f may be considered as the composition f(x) = arcsin(u(x)). Hence we use the chain rule, f '(x) = (du/dx) d(arcsin(u))/du, to differentiate function f as follows
g '(x) = (2)(1 / √(1 - u^{ 2})
= 2 / √(1 - (2x + 2)^{ 2})