Differentiate ((x²-4π²+9e^x)^n)/ln(x-1)

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Apply Quotient Rule: Step Title: Apply the Quotient Rule Concise Step Description: Since we have a function of the form f(x)/g(x), we will apply the quotient rule for differentiation, which states that the derivative of f(x)/g(x) is (g(x)f'(x) - f(x)g'(x)) / (g(x))^2. Step Calculation: Let u = (x²-4π²+9e^x)^n and v = ln(x-1). We need to find u' and v'. Step Output: Need to find u' and v'.Differentiate Numerator: Step Title: Differentiate the Numerator Concise Step Description: Differentiate u = (x²-4π²+9e^x)^n with respect to x using the chain rule. Step Calculation: u' = n(x²-4π²+9e^x)^(n-1) * (2x + 9e^x) Step Output: u' = n(x²-4π²+9e^x)^(n-1) * (2x + 9e^x)Differentiate Denominator: Step Title: Differentiate the Denominator Concise Step Description: Differentiate v = ln(x-1) with respect to x. Step Calculation: v' = 1/(x-1) Step Output: v' = 1/(x-1)Apply Quotient Rule: Step Title: Apply the Quotient Rule Concise Step Description: Substitute u, u', v, and v' into the quotient rule formula to find the derivative of the original function. Step Calculation: The derivative is (ln(x-1) * n(x²-4π²+9e^x)^(n-1) * (2x + 9e^x) - (x²-4π²+9e^x)^n * 1/(x-1)) / (ln(x-1))^2 Step Output: The derivative is (ln(x-1) * n(x²-4π²+9e^x)^(n-1) * (2x + 9e^x) - (x²-4π²+9e^x)^n * 1/(x-1)) / (ln(x-1))^2

Differentiate $\left(\frac{(x^2-4\pi^2+9e^x)^n}{\ln(x-1)}\right)$

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Differentiate ((x2−4π2+9ex)nln⁡(x−1))\left(\frac{(x^2-4\pi^2+9e^x)^n}{\ln(x-1)}\right)(ln(x−1)(x2−4π2+9ex)n​)

Differentiate $\left(\frac{(x^2-4\pi^2+9e^x)^n}{\ln(x-1)}\right)$