15–48. Derivatives Find the derivative of the following functions.
y = In (x³+1)^π

Verified step by step guidance

1. Step 1: Identify the function type. The function y = ln((x³+1)^π) is a composition of functions, specifically a logarithmic function applied to a power function.
2. Step 2: Apply the chain rule. The chain rule states that if you have a composite function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x). Here, f(u) = ln(u) and g(x) = (x³+1)^π.
3. Step 3: Differentiate the outer function. The derivative of ln(u) with respect to u is 1/u. So, f'(u) = 1/((x³+1)^π).
4. Step 4: Differentiate the inner function. The function g(x) = (x³+1)^π is a power function. Use the power rule and chain rule: g'(x) = π * (x³+1)^(π-1) * 3x².
5. Step 5: Combine the derivatives using the chain rule. Multiply the derivative of the outer function by the derivative of the inner function: y' = (1/((x³+1)^π)) * (π * (x³+1)^(π-1) * 3x²).

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the curve of the function at a specific point.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is composed of another function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions that involve nested expressions.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products or quotients of variables raised to powers. By taking the natural logarithm of both sides of the equation, we can simplify the differentiation process, especially when dealing with complex expressions. This method is particularly effective for functions involving exponentials and logarithms.