


Chapter 13 Class 11 Limits and Derivatives
Chapter 13 Class 11 Limits and Derivatives
Last updated at Dec. 16, 2024 by Teachoo
Misc 30 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₯/(π πππ π₯) Let f(x) = π₯/(π πππ π₯) Let u = x & v = sinn x β΄ f(x) = π’/π£ So, fβ(x) = (π’/π£)^β² Using quotient rule fβ(x) = (π’^β² π£ βγ π£γ^β² π’)/π£^2 Finding uβ & vβ u = x uβ = 1 Now, v = sinn x Let p = sin x v = pn By Leibnitz product rule vβ = (pn)β pβ = n pn β 1 pβ Putting p = sin x = n sinn β 1 x (sin x)β = n sinn β 1 x cos x Now, fβ(x) = (π’/π£)^β² = (π’^β² π£ βγ π£γ^β² π’)/π£^2 = ( 1 (sinπβ‘γ π₯γ ) β γπ π ππγ^(πβ1) π₯ cosβ‘γπ₯ (π₯)γ)/γγ(π ππγ^π π₯)γ^2 = ( γπ ππγ^π π₯ β π₯ (πγπ ππγ^(πβ1) π₯ cosβ‘γπ₯) γ)/γγ(π ππγ^π π₯)γ^2 = ( γπππγ^(πβπ) π . sinβ‘γπ₯ β π₯ (π γ γπ ππγ^(πβ1) π₯ cosβ‘γπ₯) γ)/γγ(π ππγ^π π₯)γ^2 = ( γπππγ^(πβπ) π γ(sinγβ‘γπ₯ β ππ₯ . γ cosβ‘γπ₯) γ)/(γπ ππγ^2π π₯) = sinβ‘γπ₯ β ππ₯ cosβ‘π₯ γ/(γπ ππγ^2π π . γπππγ^(β(πβπ) ) π) = sinβ‘γπ₯ β ππ₯ cosβ‘π₯ γ/(γπππγ^((ππ β π+π)) π) = sinβ‘γπ₯ β ππ₯ cosβ‘π₯ γ/(γπ ππγ^(π + 1) π₯) Thus, fβ(x) = πππβ‘γπ β ππ πππβ‘π γ/(γπππγ^(π + π) π)