Derivadas De Funciones Logarítmicas In 2023

Desember 21, 2023 Posting Komentar

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Are you struggling with understanding derivatives of logarithmic functions? Fear not, as we are here to break it down for you in a simplified manner. By the end of this article, you will have a clearer understanding of how to calculate derivatives of logarithmic functions.

Logarithmic Functions

Before we dive into derivatives of logarithmic functions, let's first establish what these functions are. A logarithmic function is a function that relates to the logarithm of a number. In other words, it is the inverse of an exponential function. The most common logarithmic function is the natural logarithm, denoted by ln(x).

Derivatives of Natural Logarithmic Functions

Now that we have established what logarithmic functions are, let's move on to derivatives of natural logarithmic functions. The derivative of ln(x) can be calculated using the chain rule, which states that if y = f(u) and u = g(x), then:

dy/dx = dy/du * du/dx

Using this rule, we can differentiate ln(x) as follows:

d/dx(ln(x)) = d/dx(u) * d/dx(ln(u))

In this case, u = x. Therefore, we can rewrite the expression as:

d/dx(ln(x)) = d/dx(x) * d/dx(ln(x))

As we know that the derivative of x is 1, we can simplify the expression as:

d/dx(ln(x)) = 1/x

This is the derivative of ln(x). It is important to note that the base of the logarithmic function does not affect its derivative.

Derivatives of Other Logarithmic Functions

What if we have a logarithmic function with a base other than e? In this case, we can use the change of base formula to convert it into a natural logarithmic function. The change of base formula states that:

log_a(x) = ln(x) / ln(a)

Using this formula, we can convert any logarithmic function with a base a into a natural logarithmic function. Once we have converted it, we can use the chain rule to differentiate it as we did earlier.

Examples

Let's take a look at some examples to further solidify our understanding of derivatives of logarithmic functions.

Example 1:

Find the derivative of y = ln(x²).

Solution:

Using the chain rule, we can differentiate y as follows:

dy/dx = 2x * d/dx(ln(x))

From earlier, we know that the derivative of ln(x) is 1/x. Therefore, we can substitute this value into the expression as follows:

dy/dx = 2x * 1/x

Simplifying the expression, we get:

dy/dx = 2

Therefore, the derivative of y = ln(x²) is 2.

Example 2:

Find the derivative of y = log₂(x³).

Solution:

Using the change of base formula, we can convert log₂(x³) into a natural logarithmic function as follows:

log₂(x³) = ln(x³) / ln(2)

Therefore, we can differentiate y as follows:

dy/dx = d/dx(ln(x³) / ln(2))

Using the chain rule, we can differentiate ln(x³) as follows:

d/dx(ln(x³)) = 3x²/x

Simplifying the expression, we get:

d/dx(ln(x³)) = 3

Substituting this value into the expression for dy/dx, we get:

dy/dx = 3/ln(2)

Therefore, the derivative of y = log₂(x³) is 3/ln(2).

Conclusion

Derivatives of logarithmic functions may seem daunting at first, but with the right approach, they can be easily calculated. By using the chain rule and the change of base formula, we can differentiate any logarithmic function, regardless of its base. With practice and perseverance, you can master this concept and apply it to various mathematical problems.

Remember, practice makes perfect!