Common Derivatives

Most differentiation in practice is pattern matching: memorize a short table of derivatives plus a few combination rules, and you can differentiate nearly any function that appears in statistics or disease modeling by inspection.

Reference table

Function $f(x)$	Derivative $f'(x)$
$c$ (constant)	$0$
$x^n$	$n\,x^{n-1}$
$e^x$	$e^x$
$a^x$	$a^x \ln a$
$\ln x$	$\dfrac{1}{x}$
$\log_a x$	$\dfrac{1}{x \ln a}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x = \dfrac{1}{\cos^2 x}$

Combination rules

Constant multiple:

$\frac{d}{dx}\big[c\,f(x)\big] = c\,f'(x)$

Sum (and difference):

$\frac{d}{dx}\big[f(x) \pm g(x)\big] = f'(x) \pm g'(x)$

The power rule $\frac{d}{dx}x^n = n x^{n-1}$ holds for any real $n$ , so it also covers roots ( $x^{1/2}$ ) and reciprocals ( $x^{-1}$ ).

Worked example: differentiating a polynomial

Let $f(x) = 3x^4 - 5x^2 + 7x - 2$ . Differentiate term by term using the power, constant-multiple, and sum rules:

$\begin{aligned} f'(x) &= 3 \cdot 4x^{3} - 5 \cdot 2x^{1} + 7 \cdot 1 - 0 \\ &= 12x^3 - 10x + 7 . \end{aligned}$

At $x = 1$ : $f'(1) = 12 - 10 + 7 = 9$ .

Computing it

R

D(expression(x^n), "x")        # x^n * (n * (1/x))  == n*x^(n-1)
D(expression(exp(x)), "x")     # exp(x)
D(expression(log(x)), "x")     # 1/x
D(expression(sin(x)), "x")     # cos(x)
D(expression(3*x^4 - 5*x^2 + 7*x - 2), "x")
#   3 * (4 * x^3) - 5 * (2 * x) + 7   == 12x^3 - 10x + 7

Python

import sympy as sp
x, a, n = sp.symbols("x a n")

sp.diff(x**n, x)        # n*x**n/x  == n*x**(n-1)
sp.diff(a**x, x)        # a**x*log(a)
sp.diff(sp.log(x), x)   # 1/x
sp.diff(sp.tan(x), x)   # tan(x)**2 + 1  == sec^2 x
sp.diff(3*x**4 - 5*x**2 + 7*x - 2, x)   # 12*x**3 - 10*x + 7

Julia

using Symbolics
@variables x a

Symbolics.derivative(exp(x), x)   # exp(x)
Symbolics.derivative(log(x), x)   # 1 / x
Symbolics.derivative(sin(x), x)   # cos(x)
Symbolics.derivative(3x^4 - 5x^2 + 7x - 2, x)   # 7 + 12(x^3) - 10x

Why it matters for statistics

These building blocks recur throughout biology: the exponential’s derivative governs early outbreak growth and the first-order elimination of a drug from the bloodstream, while the log’s derivative $1/x$ appears every time we differentiate a log-likelihood to fit a model. These few rules cover the derivatives you meet constantly: polynomial regression terms, the $e^x$ in exponential growth and logistic models, and the $\ln x$ at the heart of every log-likelihood. Knowing them cold lets you derive score equations and standard errors without reaching for software.