75–86. Logarithmic differentiation Use logarithmic differentiation to evaluate f'(x).
f(x) = (x+1)^3/2(x-4)^5/2 / (5x+3)^2/3

Verified step by step guidance

1. Step 1: Begin by taking the natural logarithm of both sides of the equation y = (x+1)^(3/2) * (x-4)^(5/2) / (5x+3)^(2/3). This gives us ln(y) = ln((x+1)^(3/2) * (x-4)^(5/2) / (5x+3)^(2/3)).
2. Step 2: Use the properties of logarithms to simplify the expression. The properties state that ln(a*b) = ln(a) + ln(b) and ln(a/b) = ln(a) - ln(b). Apply these to get ln(y) = (3/2)ln(x+1) + (5/2)ln(x-4) - (2/3)ln(5x+3).
3. Step 3: Differentiate both sides of the equation with respect to x. On the left side, use the chain rule to get (1/y) * dy/dx. On the right side, differentiate each term separately: (3/2)(1/(x+1)) * (d/dx)(x+1) + (5/2)(1/(x-4)) * (d/dx)(x-4) - (2/3)(1/(5x+3)) * (d/dx)(5x+3).
4. Step 4: Simplify the derivatives on the right side. The derivatives of (x+1), (x-4), and (5x+3) are 1, 1, and 5, respectively. Substitute these into the expression to get (3/2)(1/(x+1)) + (5/2)(1/(x-4)) - (2/3)(5/(5x+3)).
5. Step 5: Solve for dy/dx by multiplying both sides by y. Recall that y = (x+1)^(3/2) * (x-4)^(5/2) / (5x+3)^(2/3). Substitute this back into the expression to find dy/dx = y * [(3/2)(1/(x+1)) + (5/2)(1/(x-4)) - (2/3)(5/(5x+3))].

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate complex functions by taking the natural logarithm of both sides. This method simplifies the differentiation process, especially for products and quotients, by transforming multiplicative relationships into additive ones. It is particularly useful when dealing with functions raised to variable powers.

Product and Quotient Rules

The product and quotient rules are fundamental rules in calculus for differentiating products and quotients of functions. The product rule states that the derivative of a product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first. The quotient rule provides a similar formula for differentiating a quotient, ensuring accurate results when functions are divided.

Chain Rule

The chain rule is a key differentiation rule used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. This rule is essential for correctly differentiating functions that involve nested expressions, which is common in logarithmic differentiation.