Q. 95.0( 1 Vote )

# Differentiate with respect to if 0 < x < 1.

Answer :

Let and.

We need to differentiate u with respect to v that is find.

We have

By substituting x = tan θ, we have

[∵ sec^{2}θ – tan^{2}θ = 1]

⇒ u = sin^{–1}(2sinθcosθ)

But, sin2θ = 2sinθcosθ

⇒ u = sin^{–1}(sin2θ)

Given 0 < x < 1 ⇒ x ϵ (0, 1)

However, x = tan θ

⇒ tan θ ϵ (0, 1)

Hence, u = sin^{–1}(sin2θ) = 2θ

⇒ u = 2tan^{–1}x

On differentiating u with respect to x, we get

We know

Now, we have

By substituting x = tan θ, we have

[∵ sec^{2}θ – tan^{2}θ = 1]

⇒ v = cos^{–1}(cos^{2}θ – sin^{2}θ)

But, cos2θ = cos^{2}θ – sin^{2}θ

⇒ v = cos^{–1}(cos2θ)

However,

Hence, v = cos^{–1}(cos2θ) = 2θ

⇒ v = 2tan^{–1}x

On differentiating v with respect to x, we get

We know

We have

Thus,

Rate this question :

Show that of all the rectangles of given area, the square has the smallest perimeter.

Mathematics - Board PapersFind the value of a and b such that the following function f(x) is a continuous function:

Mathematics - Board Papers

If x sin (a + y) + sin a cos (a + y) = 0, prove that

Mathematics - ExemplarThe function f(x) = e^{|x|} is

Show that of all the rectangles with a given perimeter, the square has the largest area.

Mathematics - Board PapersIf the function f (x) given by

is continuous at x =1, find the values of a and b.

Mathematics - Board PapersFind when x and y are connected by the relation given

(x^{2} + y^{2})^{2} = xy

Let f(x) = |sinx|. Then

Mathematics - ExemplarThe derivative of cos^{-1}(2x^{2} – 1) w.r.t. cos^{-1} x is