The derivative of e^x is itself, the derivative of ln x is 1/x, and the chain rule extends both to composite functions.

What this dot point is asking

The exponential function $e^x$ is special: it is its own derivative. The natural logarithm $\ln x$ differentiates to $\tfrac1x$. These two results, combined with the chain rule, let you differentiate every exponential and logarithmic model in Stage 2.

The base results

Chain-rule extensions

When the exponent or the argument is itself a function of $x$, multiply by its derivative:

Combining with the product and quotient rules

These derivatives frequently sit inside products and quotients.

Using the log laws to simplify first

Before differentiating a complicated log, expand it with the log laws - this is far easier than the quotient or product rule.

For $y=\ln\!\left(\dfrac{x^3}{2x+1}\right)$, rewrite as $y=3\ln x-\ln(2x+1)$, then

Common errors

Why it matters

These derivatives are the basis of every exponential growth-and-decay and logarithmic application in Stage 2, and they connect directly to the antidifferentiation results in Topic 3 (since $\int e^x\,dx=e^x+C$ and $\int \tfrac1x\,dx=\ln|x|+C$ are simply these rules reversed).