Differentiate cos-1(1-x21+x2)w.r.t.tan-1x. - Mathematics and Statistics

Question

Differentiate $\cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}) w . r . t . \tan^{- 1} x .$

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Solution

Let u = $\cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}) and v = \tan^{- 1} x$ .

Then we want to find $\frac{du}{dv}$ .
Put x = tanθ.
Then = tan^–1 x.
∴ u = $\cos^{- 1} (\frac{1 - \tan^{2} θ}{1 + \tan^{2} θ})$
= cos^–1(cos2θ)
= 2θ
∴ u = 2tan^–1x
∴ $\frac{du}{dx} = 2 . \frac{d}{dx} (\tan^{- 1} x)$

= $2 \times \frac{1}{1 + x^{2}}$

= $\frac{2}{1 + x^{2}}$
Also, v =tan^–1x
∴ $\frac{dv}{dx} = \frac{d}{dx} (\tan^{- 1} x) = \frac{1}{1 + x^{2}}$

∴ $\frac{du}{dv} = \frac{(\frac{dy}{dx})}{(\frac{dv}{dx})}$

= $\frac{(\frac{2}{1 + x^{2}})}{(\frac{1}{1 + x^{2}})}$
= 2.

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Derivatives of Implicit Functions

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Q 4.4 Q 4.3 Q 4.5

Chapter 1: Differentiation - Exercise 1.4 [Page 49]

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Balbharati Mathematics and Statistics 2 (Arts and Science) [English] 12 Standard HSC Maharashtra State Board

Chapter 1 Differentiation
Exercise 1.4 | Q 4.4 | Page 49

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Differentiate cos-1(1-x21+x2)w.r.t.tan-1x. - Mathematics and Statistics

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