Question - Derivative Calculation using Logarithmic Differentiation
Solution:
To find the derivative f'(1) of the function f(x) using logarithmic differentiation, we start by taking the natural logarithm of both sides of the function.Given:f(x) = (2x - 1)^15 * (5x - 4)^5 * (9x - 8)^5 / (11x - 10)^2 * (14x - 13)^3 * (17x - 16)^3Let y = f(x), then ln(y) = ln(f(x))Taking logarithms:ln(y) = 15 * ln(2x - 1) + 5 * ln(5x - 4) + 5 * ln(9x - 8) - 2 * ln(11x - 10) - 3 * ln(14x - 13) - 3 * ln(17x - 16)Now, differentiate both sides with respect to x:1/y * dy/dx = 15/(2x - 1) * 2 + 5/(5x - 4) * 5 + 5/(9x - 8) * 9 - 2/(11x - 10) * 11 - 3/(14x - 13) * 14 - 3/(17x - 16) * 17Simplify:dy/dx = y * [ 30/(2x - 1) + 25/(5x - 4) + 45/(9x - 8) - 22/(11x - 10) - 42/(14x - 13) - 51/(17x - 16) ]Now we need to substitute x = 1 into the equation to find f'(1).First, evaluate y when x = 1, which is f(1), to substitute into the equation for dy/dx.f(1) = (2*1 - 1)^15 * (5*1 - 4)^5 * (9*1 - 8)^5 / (11*1 - 10)^2 * (14*1 - 13)^3 * (17*1 - 16)^3f(1) = (1)^15 * (1)^5 * (1)^5 / (1)^2 * (1)^3 * (1)^3f(1) = 1Now we substitute x = 1 and f(1) = 1 into the derivative equation:f'(1) = 1 * [ 30/(2*1 - 1) + 25/(5*1 - 4) + 45/(9*1 - 8) - 22/(11*1 - 10) - 42/(14*1 - 13) - 51/(17*1 - 16) ]f'(1) = [ 30/1 + 25/1 + 45/1 - 22/1 - 42/1 - 51/1 ]f'(1) = 30 + 25 + 45 - 22 - 42 - 51f'(1)= 35Therefore, using logarithmic differentiation, we find that f'(1) = 35.
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