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Misc 28 - Find derivative: x / 1 + tan x - Chapter 13 Class 11

Misc 28 - Chapter 13 Class 11 Limits and Derivatives - Part 2
Misc 28 - Chapter 13 Class 11 Limits and Derivatives - Part 3

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Misc 28 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): π‘₯/(1 + π‘‘π‘Žπ‘›β‘π‘₯ ) Let f (x) = π‘₯/(1 + π‘‘π‘Žπ‘›β‘π‘₯ ) Let u = x & v = 1 + tan x So, f(x) = 𝑒/𝑣 ∴ f’(x) = (𝑒/𝑣)^β€² Using quotient rule f’(x) = (𝑒^β€² 𝑣 βˆ’γ€– 𝑣〗^β€² 𝑒)/𝑣^2 Finding u’ & v’ u = x u’ = 1 & v = 1 + tan x v’ = (1 + tan x)’ = 0 + sec2 x = sec2 x Now, f’(x) = (𝑒/𝑣)^β€² = (𝑒^β€² 𝑣 βˆ’γ€– 𝑣〗^β€² 𝑒)/𝑣^2 = (1(1 +γ€– tan〗⁑〖π‘₯)γ€— βˆ’ 𝑠𝑒𝑐2 π‘₯ (π‘₯))/γ€–(1 +γ€– tan〗⁑〖π‘₯)γ€—γ€—^2 = (𝟏 +γ€– 𝒕𝒂𝒏〗⁑〖𝒙 βˆ’ 𝒙 π’”π’†π’„πŸ 𝒙〗)/γ€–(𝟏 +γ€– 𝒕𝒂𝒏〗⁑〖𝒙)γ€—γ€—^𝟐 = (𝑒^β€² 𝑣 βˆ’γ€– 𝑣〗^β€² 𝑒)/𝑣^2 = (1(1 +γ€– tan〗⁑〖π‘₯)γ€— βˆ’ 𝑠𝑒𝑐2 π‘₯ (π‘₯))/γ€–(1 +γ€– tan〗⁑〖π‘₯)γ€—γ€—^2 = (𝟏 +γ€– 𝒕𝒂𝒏〗⁑〖𝒙 βˆ’ 𝒙 π’”π’†π’„πŸ 𝒙〗)/γ€–(𝟏 +γ€– 𝒕𝒂𝒏〗⁑〖𝒙)γ€—γ€—^𝟐

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