Derivation Rules & Examples

Differentiation is an algorithm, not a bag of tricks. A handful of rules covers every elementary function, and they compose: autograd and hand-written backprop code are both just these rules applied in order. Knowing them cold is what lets you read a gradient formula instead of trusting it.

The idea

Two facts make the table go further. First, $\ln x$ and $e^x$ are inverses, and $1/x$ versus $e^x$ reflects that. Second, generic bases reduce to the natural ones: $a^x = e^{x \ln a}$, so $(a^x)' = a^x \ln a$, and $\log_a x = \ln x / \ln a$.

Worked example

Differentiate $f(x) = x^2 e^x$ with the product rule, taking $u = x^2$ and $v = e^x$:

At $x = 1$ this gives $1 \cdot e^1 \cdot 3 = 3e \approx 8.155$. Direct substitution into the unfactored form gives the same: $2e + e = 3e$.

Now $g(x) = (x + 1)/(x - 1)$ with the quotient rule:

At $x = 3$, $g'(3) = -2/4 = -0.5$. The derivative is negative everywhere it is defined: the function decreases on each side of its pole.

In code

import torch

x = torch.tensor(1.0, requires_grad=True)
y = x**2 * torch.exp(x)
y.backward()
print(x.grad)  # tensor(8.1548) == 3e

The chain rule, covered next, is what makes the table composable. Its job is to connect the derivative you know locally to the variable you actually care about.

Check yourself

1. Differentiate $x^3 \ln x$ using the product rule.
2. Rewrite $1/v$ as $v^{-1}$ and derive the quotient rule from the product and chain rules.
3. Differentiate $e^{2x}$ and $\ln(x^2)$ from the table plus the chain rule.

Key takeaways

• A small table of rules covers all elementary derivatives.
• Product and quotient rules are mechanical; the chain rule makes them composable.
• Reduce non-natural bases and logs to $e$ and $\ln$ before differentiating.