Derivative of ln x and log_a x formula

📚 Understanding the Derivative of $\ln x$ and $\log_a x$

Let's explore the derivatives of natural logarithms ($\ln x$) and logarithms with a base other than $e$ ($\log_a x$). These derivatives are fundamental in calculus and have various applications.

📜 History and Background

The concept of logarithms was developed by John Napier in the early 17th century as a computational aid. The natural logarithm, with base $e$, gained prominence due to its properties in calculus. The derivatives of logarithmic functions play a crucial role in solving differential equations and optimization problems.

🔑 Key Principles

• 🔍 Derivative of $\ln x$: The derivative of the natural logarithm function, $f(x) = \ln x$, is given by: $\frac{d}{dx}(\ln x) = \frac{1}{x}$. This is a fundamental result.
• 💡 Derivative of $\log_a x$: The derivative of a logarithm with base $a$, $f(x) = \log_a x$, is given by: $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$. This formula arises from the change of base formula for logarithms.
• 🔗 Change of Base: The formula for $\log_a x$ can be derived by converting the logarithm to base $e$: $\log_a x = \frac{\ln x}{\ln a}$. Since $\ln a$ is a constant, the derivative is simply $\frac{1}{\ln a} \cdot \frac{1}{x} = \frac{1}{x \ln a}$.
• 📝 Chain Rule: If we have a composite function like $\ln(u(x))$ or $\log_a(u(x))$, we need to apply the chain rule. For $\ln(u(x))$, the derivative is $\frac{u'(x)}{u(x)}$. For $\log_a(u(x))$, the derivative is $\frac{u'(x)}{u(x) \ln a}$.

➗ Examples

Here are a few examples to illustrate the application of these formulas:

1. Example 1: Find the derivative of $f(x) = \ln(x^2 + 1)$.

   Solution: Using the chain rule, $f'(x) = \frac{2x}{x^2 + 1}$.
2. Example 2: Find the derivative of $g(x) = \log_{10} x$.

   Solution: Using the formula, $g'(x) = \frac{1}{x \ln 10}$.

3. Example 3: Find the derivative of $h(x) = \log_2 (\sin x)$.

   Solution: Using the chain rule, $h'(x) = \frac{\cos x}{\sin x \ln 2} = \frac{\cot x}{\ln 2}$.

📝 Practice Quiz

Test your understanding with these practice problems:

1. Find the derivative of $y = \ln(5x)$.
2. Find the derivative of $y = \log_3(x^2)$.
3. Find the derivative of $y = x \ln x$.
4. Find the derivative of $y = \frac{\ln x}{x}$.
5. Find the derivative of $y = \ln(\cos x)$.
6. Find the derivative of $y = \log_5(2x+1)$.
7. Find the derivative of $y = (\ln x)^2$.

💡 Solutions

1. $y' = \frac{1}{x}$
2. $y' = \frac{2}{x \ln 3}$
3. $y' = \ln x + 1$
4. $y' = \frac{1 - \ln x}{x^2}$
5. $y' = -\tan x$
6. $y' = \frac{2}{(2x+1) \ln 5}$
7. $y' = \frac{2 \ln x}{x}$

🌍 Real-world Applications

• 📈 Growth and Decay Models: Logarithmic derivatives appear in models describing exponential growth or decay processes.
• 📊 Optimization Problems: They are used to find maximum or minimum values in various optimization scenarios.
• 🧪 Chemical Kinetics: In chemical kinetics, logarithmic functions are used to describe reaction rates and concentrations.

✅ Conclusion

Understanding the derivatives of $\ln x$ and $\log_a x$ is essential for calculus and its applications. By remembering the formulas and practicing with examples, you can master these important concepts. Remember to apply the chain rule when dealing with composite functions.