Primer aporte uwu
Introducción
¿Para qué sirve el cálculo?
Funciones, dominio y rango
Tipos de funciones
Cómo identificar dominio y rango de una función
Límites
El concepto de límite
Solución gráfica de los límites
Tipos de límites
Resuelve límites algebraicamente
Continuidad
La derivada
La derivada gráficamente
La definición de derivada
Obtención de derivadas utilizando la definición
Interpretando la derivada gráficamente
Derivadas de funciones algebraicas
Derivadas de una función constante y de una función con un multiplicador constante
Derivada de una función potencia
Derivada de una suma o resta de funciones
Derivada de un producto de funciones
Derivada de cociente de funciones
Derivadas de funciones trascendentes
Derivadas de funciones trigonométricas
Derivada de funciones exponenciales
Derivada de funciones logarítmicas
Bonus
Así usamos cálculo en la vida real
Regla de cadena
Qué son las funciones compuestas
Derivadas de funciones compuestas
Conclusión
Continúa con el curso de cálculo aplicado
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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.
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)} ]
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.
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.
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.
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.
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!
Contributions 7
Questions 2
Primer aporte uwu
Este video del profe Alex ayuda para, antes de ver el video, nos refresque la memoria sobre qué es un logaritmo y qué es un logaritmo neperiano (o natural).
(https://www.youtube.com/watch?v=C0BIfEB0eJM)
Dejo las fórmulas para que intenten resolver antes que el profe.
odi soy valeria :3 y gracias señor que parece periodista dando predicción del tiempo.
me volvio a colar la ultima T.T
es genial aprender y recordad las clases de matemática superior
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