Derivada de funciones logarítmicas

How to derive logarithmic functions?

Logarithmic functions, introduced by John Napier, are fundamental in mathematics because of their role as inverses of exponential functions. Deriving these functions is essential for mathematical analysis. For logarithmic functions, there are two basic rules: one for general logarithms and one for natural logarithms. Understanding these rules allows us to calculate derivatives accurately and efficiently.

What is the rule for derivatives of general logarithms?

To derive a general logarithm of base (A), we use the formula:

[ \frac{1}{x \ln(A)} ]

Where (x) is the argument of the logarithm and (\ln(A)) is the natural logarithm of base (A). For example, for the base 5 logarithm of (x), the derivative is calculated as:

[ \frac{1}{x \ln(5)} ]

How to derive natural logarithms?

The rule for deriving natural logarithms is simpler since the natural logarithm of (e) (the base of the system of natural logarithms) is 1. Therefore, the derivative of (\ln(x)) is simply:

[ \frac{1}{x} ]

This simpler result is an advantage of using natural logarithms in mathematical calculations.

Practical examples of deriving logarithmic functions

How to derive the base 5 logarithm of (x)?

We apply the above rule for general logarithms:

[ \frac{1}{x \ln(5)} ]

This example highlights the need to include the (\ln) of the base in the formula.

How to derive an expression with logarithmic quotients?

Let's consider the derivative of (\frac{"x)}{x}). This case requires the application of both the natural logarithm rule and the quotient rule:

[ F'(x) = \frac{1}{x}{x} \cdot x - \ln(x) \cdot \frac{1}{x^2} ]

Simplifying, we obtain:

[ \frac{1 - \ln(x)}{x^2} ]

We note that careful analysis of the denominator and numerator is crucial in this type of expression.

What happens when there is a product of a logarithmic function?

In the case of a function, such as (x^2 \cdot \log_{10}(x)), we use the product rule:

[ \text{Derivative = } 2x \cdot \log_{10}(x) + x^2 \cdot \frac{1}{x \ln(10)} ]

This expression can be simplified by factoring where possible. However, the development requires precision to avoid common errors of undue cancellation.

What happens when deriving logarithms of a number?

If we derive ( \log_3(6) ), we must remember that there are no variables involved, since it is a constant. The derivative of a constant is always 0.

It is important to remember that clarity and mastery of the logarithmic rules are essential to correctly solve any derivative. Practicing with various examples will help to consolidate this knowledge and avoid common confusion. Keep practicing and you will see how your understanding improves noticeably!