Misc 6 - Find derivative: 1 + 1/x / 1 - 1/x - Chapter 13 CBSE - Derivatives by formula - x^n formula

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Misc 6 Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): 1 + 1﷮𝑥﷯﷮1 − 1﷮𝑥﷯﷯ Let f(x) = 1 + 1﷮𝑥﷯﷮1 − 1﷮𝑥﷯﷯ = x + 1﷮x﷯﷮ x − 1﷮x﷯﷯ = x + 1﷮x﷯ × 𝑥﷮𝑥 − 1﷯ = 𝑥 + 1﷮𝑥 − 1 ﷯ So, f(x) = 𝑥 + 1﷮𝑥 − 1 ﷯ ∴ f(x) = 𝑢﷮𝑣﷯ So, f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ = u﷮′﷯v − v﷮′﷯u﷮ v﷮2﷯﷯ Now, u = x + 1 u’ = 1.x1–1 + 0 = 1.x0 = 1 v = x – 1 v’ = 1.x1–1 – 0 = 1.x0 = 1 Now, f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = 1 x − 1﷯ −1 x + 1﷯ ﷮ x − 1﷯2﷯ = x − 1 − x − 1﷮ x − 1﷯2﷯ = − 2﷮ x − 1﷯2﷯ Hence, f’ (x) = − 𝟐﷮ 𝐱 − 𝟏﷯𝟐﷯

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