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Derivatives of composite, implicit and inverse trigonometric functions

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Derivatives of composite, implicit and inverse trigonometric functions - Lesson Summary

Derivatives of Composite, Implicit and Inverse Trigonometric Functions.

Chain rule:

f is a real valued function, which is a composite of two functions, u and v; i.e. f = u • v. Suppose t = v(x) and if dt/dx and du/dt exist, then: df/dx = du/dt • dt/dx

The chain rule can be also be applied to functions that are composites of more than two functions.

f is a real valued function, which is a composite of three functions, u, v and w; i.e. f = (u • v) • w. If s = w(x) and t = v(s), then:


df/dx = d(u • v)/ds • ds/dx = du//dt • dt/ds • ds/dx

Exmple :

f(s) = (x - 3)50

Sol:

Let f(x) = (h • g)(x) Where g(x) = x - 3 and h(x) = x50

Suppose t = x - 3 ⇒ f(x) = h(t) = t50

df/dx = dh/dt • dt/dx

= d(t50)/dt • d(x - 3)/dx

= 50t49 × 1 = 50t49 Substitute the value of t.

∴ df/dx = 50(x - 3)49

An expression involving x and y that is easily solved for y and written in the form y = f(x) is said to be y is given as an explicit function of x.

Ex: y = f(x) = 2x + 5

A function that is formed implicitly is called an implicit function.

y - 2x = 5

Inverse Trigonometric Functions

     f(x)

    sin-1 x

    cos-1 x

   tan-1 x

   cot-1 x

    sec-1 x

   cosex-1 x

    f ' (x)

   1/√(1-x2)

   -1/√(1-x2)

    1/1+x2

  -1/1+x2

  1/(x√(x2 - 1))

   -1/(x√(x2 - 1))

Domain of f '

    (-1, 1)

    (-1, 1)

     R

    R

 (-∞, -1) ∪ (1, ∞)

 (-∞, -1) ∪ (1, ∞)

Ex: f(x) = sin-1x

Let y = sin-1x

⇒ x = sin y ..... (1)

dx/dx = d(sin y)/dx ⇒ 1 = cos y dy/dx

⇒ dy/dx = 1/cos y

= 1/√(1 - sin2 y) = 1/√(1 - sin y)2 [ '.' cos2y + sin2 y = 1]

= 1/ √(1 - x2) [From (1)]

∴ dy/dx = 1/√(1 - x2)

Note:

The derivative of sin-1 x is only possible for x ∈ (-1, 1)

Similarly we can find all the derivatives of inverse trigonometric functions.

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