Question

Differentiate y = tan - 1 _a - x 1 + ax DIFFERENTIATION Put a = tana and x = tan 0. tana - tane tan + tana tane + y = tan = tan = α - 0 tan (α - 0) = a - tan X. Here a is a constant since a is a constant. dy 1 dx (1 + x²) (B.Sc. 1989)​

Answer 1

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$let \: y = {tan}^{ - 1} ( \frac{a + x}{1 - ax} ) \\ putting \: a = tan \alpha \: \: and \: \: x = tan \theta \\ = > y = {tan}^{ - 1} ( \frac{tan \alpha +tan \theta }{ 1-tan \alpha.tan \theta } ) \\ = {tan}^{ - 1} (tan( \alpha + \theta)) \\ = \alpha + \theta \\ so , \: \: \: \: \: \: \: \: y = {tan}^{ - 1} \alpha + {tan}^{ - 1} x \\ differentiating \: w.r.t.x \\ ∴\frac{dy}{dx} = \frac{d}{dx} ( {tan}^{ - 1} \alpha + {tan}^{ - 1}x ) \\ ∴ \frac{dy}{dx} = 0 + \frac{1}{1 + {x}^{2} } \\ ∴\frac{dy}{dx} = \frac{1}{1 + {x}^{2} }$