๐ Understanding ln(x) vs. ln(g(x)) with the Chain Rule
Let's demystify the difference between differentiating $ln(x)$ and $ln(g(x))$, especially when the chain rule comes into play. It's all about that inner function!
Definition of ln(x)
$ln(x)$ represents the natural logarithm of $x$. Its derivative is straightforward:
$\frac{d}{dx} ln(x) = \frac{1}{x}$
Definition of ln(g(x))
$ln(g(x))$ represents the natural logarithm of a function $g(x)$. Here, $g(x)$ is the 'inner' function. Differentiating this requires the chain rule:
$\frac{d}{dx} ln(g(x)) = \frac{1}{g(x)} * g'(x) = \frac{g'(x)}{g(x)}$
๐ Comparison Table: ln(x) vs. ln(g(x))
| Feature |
ln(x) |
ln(g(x)) |
| Definition |
Natural logarithm of x |
Natural logarithm of a function g(x) |
| Derivative |
$\frac{1}{x}$ |
$\frac{g'(x)}{g(x)}$ |
| Chain Rule Needed? |
No |
Yes |
| Example |
$\frac{d}{dx} ln(x) = \frac{1}{x}$ |
If $g(x) = x^2$, then $\frac{d}{dx} ln(x^2) = \frac{2x}{x^2} = \frac{2}{x}$ |
๐ก Key Takeaways
- ๐ Basic Natural Logarithm: $ln(x)$ is the fundamental natural logarithm with a simple derivative.
- ๐งช Chain Rule Application: $ln(g(x))$ requires the chain rule because you're taking the logarithm of a function, not just a variable.
- ๐ Derivative of the Inner Function: Remember to multiply by the derivative of the inner function, $g'(x)$, when differentiating $ln(g(x))$.
- ๐ง Simplification: After applying the chain rule, simplify the expression if possible.
- ๐ Common Mistake: Forgetting to apply the chain rule to $ln(g(x))$. Always identify the inner function first!