New Page 2

3)
9(1/2)(y')(2/y)sin x + 9(ln y/2)cos x = e^xtan 2y + e^x(sec² 2y)2y'
y'(9/y)sin x - y'2e^x(sec² 2y) = e^xtan 2y - 9(ln y/2)cos x
y'[(9/y)sin x - 2e^x(sec² 2y)] = e^xtan 2y - 9(ln y/2)cos x
y' = [e^xtan 2y - 9(ln y/2)cos x]/[(9/y)sin x - 2e^x(sec² 2y)]

4)
ln y = 4xsin x ln (sin x)
y’/y = 4sin x ln (sin x) + 4xcos x ln (sin x) + 4xcos x
y’ = (sin x)^{4xln x}[4sin x ln (sin x) + 4xcos x ln (sin x) + 4xcos x]

5)
y’ = e^x + xe^x

6)
ln y = (4x + 2)ln (x² – 6x)
y’/y = 4 ln (x² – 6x) + (4x + 2)(2x – 6)/(x² – 6x)
y’ = (x² – 6x)^{4x + 2}[4 ln (x² – 6x) + (4x + 2)(2x – 6)/(x² – 6x)]

7)
ln y = xln (2x⁴ + e^x)
y’/y = ln (2x⁴ + e^x) + x(8x³ + e^x)/(2x⁴ + e^x)
y’ = (2x⁴ + e^x)^x[ln (2x⁴ + e^x) + x(8x³ + e^x)/(2x⁴ + e^x)]

8)
ln y = tan x ln (tan x)
y’/y = sec² x ln (tan x) + sec² x
y' = (tan x)^{tan x}[sec² x ln (tan x) + sec² x]

9)
ln y = sin x ln (ln 4x)
y’/y = cos x ln (ln 4x) + (sin x)/(xln 4x)
y’ = ycos x ln (ln 4x) + y(sin x)/(xln 4x)
y’ = (ln 4x)^{sin x}cos x ln (ln 4x) + (ln 4x)^{sin x}(sin x)/(xln 4x)

Answers – Advanced Differentiation