In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
35. y = ln((x²+1)^5/√(1-x))

Verified step by step guidance

1. Rewrite the function to simplify differentiation by expressing the logarithm of a quotient and powers as sums and differences: use the property \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\) and \(\ln(a^n) = n \ln(a)\). So rewrite \(y = \ln\left(\frac{(x^2+1)^5}{\sqrt{1-x}}\right)\) as \(y = 5 \ln(x^2 + 1) - \frac{1}{2} \ln(1 - x)\).
2. Differentiate each term separately with respect to \(x\) using the chain rule. For the first term, \(5 \ln(x^2 + 1)\), the derivative is \(5 \cdot \frac{1}{x^2 + 1} \cdot \frac{d}{dx}(x^2 + 1)\).
3. Calculate the derivative inside the chain rule for the first term: \(\frac{d}{dx}(x^2 + 1) = 2x\).
4. For the second term, \(-\frac{1}{2} \ln(1 - x)\), differentiate using the chain rule: \(-\frac{1}{2} \cdot \frac{1}{1 - x} \cdot \frac{d}{dx}(1 - x)\).
5. Calculate the derivative inside the chain rule for the second term: \(\frac{d}{dx}(1 - x) = -1\). Then combine all parts to write the derivative \(\frac{dy}{dx}\) as the sum of these differentiated terms.

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation involves taking the natural logarithm of a function to simplify the differentiation process, especially when the function is a product, quotient, or power. It transforms complicated expressions into sums, differences, and products, making derivatives easier to compute.

Chain Rule

The chain rule is used to differentiate composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This is essential when differentiating functions like ln(u(x)) or powers of functions.

Properties of Logarithms

Properties of logarithms, such as ln(a^b) = b ln(a) and ln(a/b) = ln(a) - ln(b), allow simplification of complex logarithmic expressions. Applying these properties helps break down the given function into simpler terms before differentiation.