f(x)=log_(5)((2x-4)/(x+3)-5)

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Understand Logarithmic Function: Understand the function and the base of the logarithm. The function f(x) = log_5((2x-4)/(x+3)-5) is a logarithmic function with base 5. The argument of the logarithm is a rational function minus 5. We need to find the derivative of this function with respect to x.Apply Chain Rule for Differentiation: Apply the chain rule for logarithmic differentiation. The derivative of a logarithm with base a (where a > 0 and a ≠ 1) is given by the formula d/dx(log_a(u)) = (1/u) * (du/dx) * (1/ln(a)), where u is a function of x. In this case, u = (2x-4)/(x+3)-5.Differentiate Argument of Logarithm: Differentiate the argument of the logarithm. We need to find the derivative of u = (2x-4)/(x+3)-5 with respect to x. This requires the quotient rule and the constant rule. The quotient rule is d/dx(v/w) = (w*dv/dx - v*dw/dx) / w^2, where v and w are functions of x. Let's differentiate the numerator v = 2x - 4 and the denominator w = x + 3. dv/dx = d/dx(2x - 4) = 2 dw/dx = d/dx(x + 3) = 1 Now apply the quotient rule: du/dx = ((x+3)*2 - (2x-4)*1) / (x+3)^2 du/dx = (2x+6 - 2x+4) / (x+3)^2 du/dx = (10) / (x+3)^2Combine Results for Derivative: Combine the results to find the derivative of the original function. Now we can use the result from Step 2 and Step 3 to find the derivative of f(x): f'(x) = (1/((2x-4)/(x+3)-5)) * (10 / (x+3)^2) * (1/ln(5))Simplify Expression: Simplify the expression. We can simplify the expression by multiplying the terms: f'(x) = 10 / (((2x-4)/(x+3)-5) * (x+3)^2 * ln(5))Check for Errors: Check for any possible simplifications or errors. Upon reviewing the steps, there do not appear to be any mathematical errors. The expression for the derivative seems to be as simplified as possible given the complexity of the original function.

$f(x)=\log_{5}\left(\frac{2x-4}{x+3}-5\right)$

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$f(x)=\log_{5}\left(\frac{2x-4}{x+3}-5\right)$