Calculate the derivative of the following functions. In some cases, it is useful to use the properties of logarithms to simplify the functions before computing f'(x).
f(x) = In(3x + 1)⁴

Verified step by step guidance

1. First, recognize that the function f(x) = ln((3x + 1)^4) can be simplified using the properties of logarithms. Specifically, use the property ln(a^b) = b * ln(a) to rewrite the function as f(x) = 4 * ln(3x + 1).
2. Next, apply the derivative rule for a constant multiplied by a function. The derivative of a constant times a function, c * g(x), is c * g'(x). Here, the constant is 4, and the function is ln(3x + 1).
3. Now, find the derivative of ln(3x + 1). Recall that the derivative of ln(u) with respect to x is (1/u) * (du/dx). In this case, u = 3x + 1, so you need to find du/dx.
4. Calculate du/dx for u = 3x + 1. The derivative of 3x + 1 with respect to x is simply 3, since the derivative of x is 1 and the derivative of a constant is 0.
5. Combine the results: the derivative of ln(3x + 1) is (1/(3x + 1)) * 3. Therefore, the derivative of f(x) = 4 * ln(3x + 1) is 4 * (3/(3x + 1)).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve of the function at any given point. The derivative is denoted as f'(x) and can be calculated using various rules, such as the power rule, product rule, and chain rule.

Logarithmic Properties

Logarithmic properties are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (log(a*b) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^b) = b*log(a)). These properties are particularly useful in calculus for simplifying functions before differentiation, making the derivative easier to compute.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is essential when dealing with functions that involve nested expressions, such as logarithmic or exponential functions.