Calculus Help!!!!

Question

Use logarithmic differentiation to find a formula for f'(x) if f(x) = x^sin(x) * (x-1)^20 * (3x^2 + 2)^24.

Answer 1

f(x) = x^sin(x) • (x-1)²⁰ • (3x² + 2)²⁴

y = x^sin(x) • (x-1)²⁰ • (3x² + 2)²⁴

ln(y) = ln(x^sin(x) • (x-1)²⁰ • (3x² + 2)²⁴)

ln(y) = ln(x^sin(x)) + ln((x-1)²⁰) + ln((3x² + 2)²⁴)

ln(y) = (sin(x))ln(x) + 20ln(x-1) + 24ln(3x² + 2)

Take the derivative implicitly:

(ln(y))' = [(sin(x))ln(x)]' + (20ln(x-1))' + (24ln(3x² + 2))'

(1/y)•y' = [cos(x)ln(x) + sin(x)(1/x)] + 20/(x - 1) + 24/(3x² + 2)•(6x)

(1/y)•y' = cos(x)ln(x) + sin(x)/x + 20/(x - 1) + 144x/(3x² + 2)

y' = y[cos(x)ln(x) + sin(x)/x + 20/(x - 1) + 144x/(3x² + 2)]

If you want this only in terms of "x" you can replace the "y" with "x^sin(x) • (x-1)²⁰ • (3x² + 2)²⁴"