91Ó°ÊÓ

Derivatives are a key concept in calculus, helping us understand how functions change. One important type of function is the logarithmic function. The natural logarithm, denoted as \( \ln x \), is a common type that appears frequently. Knowing how to differentiate it is crucial. The derivative of \( \ln x \) is \( \frac{1}{x} \).

This means that for any value of \( x \), the rate at which \( \ln x \) changes is driven by the reciprocal of \( x \). If you encounter more complicated expressions involving logarithms, you often find yourself working with this basic rule, repeatedly.

For functions like \( g(x) = 12 - 7 \ln x \), we apply the derivative of the logarithmic part separately. Always remember, when differentiating, you handle each part of the sum individually.

The constant rule in derivatives is simple but extremely useful. This rule states that the derivative of a constant is zero. Why? Because a constant does not change, and if something doesn't change, its rate of change is zero.

In the given function \( g(x) = 12 - 7 \ln x \), the constant part is \( 12 \). When differentiating \( 12 \), you simply get zero: \( \frac{d}{dx}(12) = 0 \).

The importance of the constant rule becomes especially evident in functions that combine constants with variables, where it simplifies calculations considerably by eliminating terms outright.

The constant multiple rule is another vital concept when differentiating functions. It allows you to differentiate expressions that are a constant multiplied by a function.

The rule says that if you have a term like \( c \cdot f(x) \), where \( c \) is a constant, the derivative is \( c \cdot f'(x) \). This is straightforward but powerful because it lets you pull constants out of the differentiation process, focusing on the variable part instead.

For the term \( -7 \ln x \) in the function \( g(x) = 12 - 7 \ln x \), the constant is \( -7 \). Applying the constant multiple rule, you multiply \( -7 \) by the derivative of \( \ln x \), resulting in \( -7 \cdot \frac{1}{x} \), which simplifies to \( -\frac{7}{x} \).

This rule is a time-saver and reduces complexity, especially as you tackle longer and more intricate functions.