Derivative Calculation using Logarithmic Differentiation
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)^3
Let 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) * 17
Simplify:
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)^3
f(1) = (1)^15 * (1)^5 * (1)^5 / (1)^2 * (1)^3 * (1)^3
f(1) = 1
Now 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 - 51
f'(1)= 35
Therefore, using logarithmic differentiation, we find that f'(1) = 35.
