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Derivatives of Inverse Trigonometric Functions

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 / sqrt(1 - x 2)


2 - Derivative of arccos x.

The derivative of f(x) = arccos x is given by

f '(x) = - 1 / sqrt(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 sqrt(x 2 - 1))


6 - Derivative of arccsc x.

The derivative of f(x) = arccsc x is given by

f '(x) = - 1 / (x sqrt(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 / sqrt(1 - x 2)) + arcsin x * 1 = x / sqrt(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 / sqrt(1 - u 2)

    = 2 / sqrt(1 - (2x + 2) 2)



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Updated: 2 April 2013

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