Quick Guide: Calculating Derivatives of ln(g(x)) Using the Chain Rule

📚 Calculating Derivatives of ln(g(x)) with the Chain Rule: A Comprehensive Guide

The derivative of the natural logarithm of a function, denoted as $ln(g(x))$, is a common topic in calculus. Mastering this skill requires a solid understanding of the chain rule and the derivative of the natural logarithm function. This guide breaks down the concept, its historical roots, key principles, and provides real-world examples.

📜 History and Background

The concept of derivatives dates back to the 17th century with the independent work of Isaac Newton and Gottfried Wilhelm Leibniz. The natural logarithm, with base $e$ (Euler's number), plays a crucial role in calculus and mathematical analysis. The chain rule is essential for differentiating composite functions, making it indispensable when dealing with $ln(g(x))$.

🔑 Key Principles

• 🧮 The Chain Rule: This rule states that the derivative of a composite function $f(g(x))$ is $f'(g(x)) * g'(x)$. In simpler terms, you take the derivative of the outer function evaluated at the inner function, then multiply by the derivative of the inner function.
• 🌱 Derivative of ln(x): The derivative of the natural logarithm function, $ln(x)$, is $1/x$. This is a fundamental rule in calculus.
• 💡 Applying the Chain Rule to ln(g(x)): When you have $ln(g(x))$, the outer function is $ln(x)$ and the inner function is $g(x)$. Applying the chain rule, the derivative of $ln(g(x))$ is $\frac{1}{g(x)} * g'(x)$, which simplifies to $\frac{g'(x)}{g(x)}$.

📝 Step-by-Step Calculation

To calculate the derivative of $ln(g(x))$:

1. Identify the inner function $g(x)$.
2. Find the derivative of the inner function, $g'(x)$.
3. Divide the derivative of the inner function by the original inner function: $\frac{g'(x)}{g(x)}$.

➕ Real-World Examples

Example 1:

Find the derivative of $ln(x^2 + 1)$

1. $g(x) = x^2 + 1$
2. $g'(x) = 2x$
3. $\frac{g'(x)}{g(x)} = \frac{2x}{x^2 + 1}$

Example 2:

Find the derivative of $ln(sin(x))$

1. $g(x) = sin(x)$
2. $g'(x) = cos(x)$
3. $\frac{g'(x)}{g(x)} = \frac{cos(x)}{sin(x)} = cot(x)$

Example 3:

Find the derivative of $ln(e^x)$

1. $g(x) = e^x$
2. $g'(x) = e^x$
3. $\frac{g'(x)}{g(x)} = \frac{e^x}{e^x} = 1$

✍️ Practice Quiz

Find the derivatives of the following functions:

1. $ln(x^3)$
2. $ln(cos(x))$
3. $ln(2x + 5)$

✔️ Solutions

1. $\frac{3x^2}{x^3} = \frac{3}{x}$
2. $\frac{-sin(x)}{cos(x)} = -tan(x)$
3. $\frac{2}{2x + 5}$

🎯 Conclusion

Calculating derivatives of $ln(g(x))$ using the chain rule involves understanding the chain rule itself and the derivative of the natural logarithm. By following the steps outlined and practicing with examples, you can master this important calculus technique. Remember to identify the inner function, find its derivative, and then apply the formula $\frac{g'(x)}{g(x)}$.