Linear Approximation Cheatsheet

1. Definition

When $x$ is close to some reference value $x_0$ (in other words $\Delta x \approx 0$ ), you can approximate $f(x)$ as the tangent line at that reference value.

x_0

0.30

\Delta x

0.60

$f(x_0)$: 1.349859
$\tilde f(x)$ on the tangent line: 2.159774
$f(x)$ on the curve: 2.459603
error ( $\left| f(x) - \tilde f(x) \right|$ ): 0.299829

form	formula	analog
point–slope form	$f(x) \approx f(x_0) + f'(x_0)(x - x_0)$	$y - y_0 = m(x - x_0)$
slope–intercept form	$f(x + \Delta x) \approx f'(x)\,\Delta x + f(x)$	$y = mx + b$

2. Cheatsheet

Each entry gives you $f$ and $f'$ , then lets you pick the reference point. The formula, the table and the graph all follow that choice. Pick any $x_0$ to see the general form and drag the reference around.

2.1 Trigonometric

2.1.1 $\sin x$

\sin x

\approx

\sin x_0 + \cos x_0\,(x - x_0)

reference

0

\frac{\pi}{6}

\frac{\pi}{4}

\frac{\pi}{2}

0

\sin x

\approx

\sin 0 + \cos 0\,(x - 0)

\approx

x

	$f(x)$	$\tilde f(x)$	error
$0 + 0$	$0.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$0.010000$	$0.010000$	$1.67 \times 10^{-7}$
$0 + \frac{1}{10}$	$0.099833$	$0.100000$	$0.000167$
$0 + \frac{1}{2}$	$0.479426$	$0.500000$	$0.020574$
$0 + 1$	$0.841471$	$1.000000$	$0.158529$

\sin x

\tilde f

\frac{\pi}{6}

\sin x

\approx

\sin\frac{\pi}{6} + \cos\frac{\pi}{6}\left(x - \frac{\pi}{6}\right)

\approx

\frac{1}{2} + \frac{\sqrt{3}}{2}\left(x - \frac{\pi}{6}\right)

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{6} + 0$	$0.500000$	$0.500000$	$0.000000$
$\frac{\pi}{6} + \frac{1}{100}$	$0.508635$	$0.508660$	$2.51 \times 10^{-5}$
$\frac{\pi}{6} + \frac{1}{10}$	$0.583960$	$0.586603$	$0.002642$
$\frac{\pi}{6} + \frac{1}{2}$	$0.853986$	$0.933013$	$0.079027$
$\frac{\pi}{6} + 1$	$0.998886$	$1.366025$	$0.367139$

\sin x

\tilde f

\frac{\pi}{4}

\sin x

\approx

\sin\frac{\pi}{4} + \cos\frac{\pi}{4}\left(x - \frac{\pi}{4}\right)

\approx

\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}\left(x - \frac{\pi}{4}\right)

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{4} + 0$	$0.707107$	$0.707107$	$0.000000$
$\frac{\pi}{4} + \frac{1}{100}$	$0.714142$	$0.714178$	$3.55 \times 10^{-5}$
$\frac{\pi}{4} + \frac{1}{10}$	$0.774167$	$0.777817$	$0.003650$
$\frac{\pi}{4} + \frac{1}{2}$	$0.959550$	$1.060660$	$0.101111$
$\frac{\pi}{4} + 1$	$0.977061$	$1.414214$	$0.437152$

\sin x

\tilde f

\frac{\pi}{2}

\sin x

\approx

\sin\frac{\pi}{2} + \cos\frac{\pi}{2}\left(x - \frac{\pi}{2}\right)

\approx

1

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{2} + 0$	$1.000000$	$1.000000$	$0.000000$
$\frac{\pi}{2} + \frac{1}{100}$	$0.999950$	$1.000000$	$5.00 \times 10^{-5}$
$\frac{\pi}{2} + \frac{1}{10}$	$0.995004$	$1.000000$	$0.004996$
$\frac{\pi}{2} + \frac{1}{2}$	$0.877583$	$1.000000$	$0.122417$
$\frac{\pi}{2} + 1$	$0.540302$	$1.000000$	$0.459698$

\sin x

\tilde f

At 0 this is the small-angle rule. At π/2 the curve is flat, so the approximation is just the constant 1.

2.1.2 $\cos x$

\cos x

\approx

\cos x_0 - \sin x_0\,(x - x_0)

reference

0

\frac{\pi}{6}

\frac{\pi}{3}

\frac{\pi}{2}

0

\cos x

\approx

\cos 0 - \sin 0\,(x - 0)

\approx

1

	$f(x)$	$\tilde f(x)$	error
$0 + 0$	$1.000000$	$1.000000$	$0.000000$
$0 + \frac{1}{100}$	$0.999950$	$1.000000$	$5.00 \times 10^{-5}$
$0 + \frac{1}{10}$	$0.995004$	$1.000000$	$0.004996$
$0 + \frac{1}{2}$	$0.877583$	$1.000000$	$0.122417$
$0 + 1$	$0.540302$	$1.000000$	$0.459698$

\cos x

\tilde f

\frac{\pi}{6}

\cos x

\approx

\cos\frac{\pi}{6} - \sin\frac{\pi}{6}\left(x - \frac{\pi}{6}\right)

\approx

\frac{\sqrt{3}}{2} - \frac{1}{2}\left(x - \frac{\pi}{6}\right)

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{6} + 0$	$0.866025$	$0.866025$	$0.000000$
$\frac{\pi}{6} + \frac{1}{100}$	$0.860982$	$0.861025$	$4.32 \times 10^{-5}$
$\frac{\pi}{6} + \frac{1}{10}$	$0.811782$	$0.816025$	$0.004243$
$\frac{\pi}{6} + \frac{1}{2}$	$0.520296$	$0.616025$	$0.095729$
$\frac{\pi}{6} + 1$	$0.047180$	$0.366025$	$0.318845$

\cos x

\tilde f

\frac{\pi}{3}

\cos x

\approx

\cos\frac{\pi}{3} - \sin\frac{\pi}{3}\left(x - \frac{\pi}{3}\right)

\approx

\frac{1}{2} - \frac{\sqrt{3}}{2}\left(x - \frac{\pi}{3}\right)

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{3} + 0$	$0.500000$	$0.500000$	$0.000000$
$\frac{\pi}{3} + \frac{1}{100}$	$0.491315$	$0.491340$	$2.49 \times 10^{-5}$
$\frac{\pi}{3} + \frac{1}{10}$	$0.411044$	$0.413397$	$0.002354$
$\frac{\pi}{3} + \frac{1}{2}$	$0.023597$	$0.066987$	$0.043391$
$\frac{\pi}{3} + 1$	$-0.458584$	$-0.366025$	$0.092559$

\cos x

\tilde f

\frac{\pi}{2}

\cos x

\approx

\cos\frac{\pi}{2} - \sin\frac{\pi}{2}\left(x - \frac{\pi}{2}\right)

\approx

\frac{\pi}{2} - x

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{2} + 0$	$6.12 \times 10^{-17}$	$6.12 \times 10^{-17}$	$0.000000$
$\frac{\pi}{2} + \frac{1}{100}$	$-0.010000$	$-0.010000$	$1.67 \times 10^{-7}$
$\frac{\pi}{2} + \frac{1}{10}$	$-0.099833$	$-0.100000$	$0.000167$
$\frac{\pi}{2} + \frac{1}{2}$	$-0.479426$	$-0.500000$	$0.020574$
$\frac{\pi}{2} + 1$	$-0.841471$	$-1.000000$	$0.158529$

\cos x

\tilde f

Flat at 0, so the line there is the constant 1. At π/2 it is steepest and the approximation is a pure slope.

2.1.3 $\tan x$

\tan x

\approx

\tan x_0 + \sec^2 x_0\,(x - x_0)

reference

0

\frac{\pi}{6}

\frac{\pi}{4}

0

\tan x

\approx

\tan 0 + \sec^2 0\,(x - 0)

\approx

x

	$f(x)$	$\tilde f(x)$	error
$0 + 0$	$0.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$0.010000$	$0.010000$	$3.33 \times 10^{-7}$
$0 + \frac{1}{10}$	$0.100335$	$0.100000$	$0.000335$
$0 + \frac{1}{2}$	$0.546302$	$0.500000$	$0.046302$
$0 + 1$	$1.557408$	$1.000000$	$0.557408$

\tan x

\tilde f

\frac{\pi}{6}

\tan x

\approx

\tan\frac{\pi}{6} + \sec^2\frac{\pi}{6}\left(x - \frac{\pi}{6}\right)

\approx

\frac{1}{\sqrt{3}} + \frac{4}{3}\left(x - \frac{\pi}{6}\right)

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{6} + 0$	$0.577350$	$0.577350$	$0.000000$
$\frac{\pi}{6} + \frac{1}{100}$	$0.590761$	$0.590684$	$7.79 \times 10^{-5}$
$\frac{\pi}{6} + \frac{1}{10}$	$0.719356$	$0.710684$	$0.008672$
$\frac{\pi}{6} + \frac{1}{2}$	$1.641346$	$1.244017$	$0.397329$
$\frac{\pi}{6} + 1$	$21.171805$	$1.910684$	$19.261121$

\tan x

\tilde f

\frac{\pi}{4}

\tan x

\approx

\tan\frac{\pi}{4} + \sec^2\frac{\pi}{4}\left(x - \frac{\pi}{4}\right)

\approx

1 + 2\left(x - \frac{\pi}{4}\right)

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{4} + 0$	$1.000000$	$1.000000$	$0.000000$
$\frac{\pi}{4} + \frac{1}{100}$	$1.020203$	$1.020000$	$0.000203$
$\frac{\pi}{4} + \frac{1}{10}$	$1.223049$	$1.200000$	$0.023049$
$\frac{\pi}{4} + \frac{1}{2}$	$3.408223$	$2.000000$	$1.408223$
$\frac{\pi}{4} + 1$	$-4.588038$	$3.000000$	$7.588038$

\tan x

\tilde f

Near 0, tan x flattens to x, the same as sin x. It steepens fast as you approach π/2.

2.1.4 $\sec x$

\sec x

\approx

\sec x_0 + \sec x_0 \tan x_0\,(x - x_0)

reference

0

\frac{\pi}{6}

\frac{\pi}{4}

0

\sec x

\approx

\sec 0 + \sec 0\tan 0\,(x - 0)

\approx

1

	$f(x)$	$\tilde f(x)$	error
$0 + 0$	$1.000000$	$1.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.000050$	$1.000000$	$5.00 \times 10^{-5}$
$0 + \frac{1}{10}$	$1.005021$	$1.000000$	$0.005021$
$0 + \frac{1}{2}$	$1.139494$	$1.000000$	$0.139494$
$0 + 1$	$1.850816$	$1.000000$	$0.850816$

\sec x

\tilde f

\frac{\pi}{6}

\sec x

\approx

\sec\frac{\pi}{6} + \sec\frac{\pi}{6}\tan\frac{\pi}{6}\left(x - \frac{\pi}{6}\right)

\approx

\frac{2}{\sqrt{3}} + \frac{2}{3}\left(x - \frac{\pi}{6}\right)

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{6} + 0$	$1.154701$	$1.154701$	$0.000000$
$\frac{\pi}{6} + \frac{1}{100}$	$1.161464$	$1.161367$	$9.70 \times 10^{-5}$
$\frac{\pi}{6} + \frac{1}{10}$	$1.231858$	$1.221367$	$0.010490$
$\frac{\pi}{6} + \frac{1}{2}$	$1.921983$	$1.488034$	$0.433949$
$\frac{\pi}{6} + 1$	$21.195408$	$1.821367$	$19.374041$

\sec x

\tilde f

\frac{\pi}{4}

\sec x

\approx

\sec\frac{\pi}{4} + \sec\frac{\pi}{4}\tan\frac{\pi}{4}\left(x - \frac{\pi}{4}\right)

\approx

\sqrt{2} + \sqrt{2}\left(x - \frac{\pi}{4}\right)

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{4} + 0$	$1.414214$	$1.414214$	$0.000000$
$\frac{\pi}{4} + \frac{1}{100}$	$1.428570$	$1.428356$	$0.000215$
$\frac{\pi}{4} + \frac{1}{10}$	$1.579825$	$1.555635$	$0.024191$
$\frac{\pi}{4} + \frac{1}{2}$	$3.551899$	$2.121320$	$1.430579$
$\frac{\pi}{4} + 1$	$-4.695752$	$2.828427$	$7.524180$

\sec x

\tilde f

Flat at 0 like cos, so the line there is the constant 1.

2.1.5 $\cot x$

\cot x

\approx

\cot x_0 - \csc^2 x_0\,(x - x_0)

reference

\frac{\pi}{4}

\frac{\pi}{2}

\frac{\pi}{4}

\cot x

\approx

\cot\frac{\pi}{4} - \csc^2\frac{\pi}{4}\left(x - \frac{\pi}{4}\right)

\approx

1 - 2\left(x - \frac{\pi}{4}\right)

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{4} + 0$	$1.000000$	$1.000000$	$0.000000$
$\frac{\pi}{4} + \frac{1}{100}$	$0.980197$	$0.980000$	$0.000197$
$\frac{\pi}{4} + \frac{1}{10}$	$0.817629$	$0.800000$	$0.017629$
$\frac{\pi}{4} + \frac{1}{2}$	$0.293408$	$0.000000$	$0.293408$
$\frac{\pi}{4} + 1$	$-0.217958$	$-1.000000$	$0.782042$

\cot x

\tilde f

\frac{\pi}{2}

\cot x

\approx

\cot\frac{\pi}{2} - \csc^2\frac{\pi}{2}\left(x - \frac{\pi}{2}\right)

\approx

\frac{\pi}{2} - x

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{2} + 0$	$6.12 \times 10^{-17}$	$6.12 \times 10^{-17}$	$0.000000$
$\frac{\pi}{2} + \frac{1}{100}$	$-0.010000$	$-0.010000$	$3.33 \times 10^{-7}$
$\frac{\pi}{2} + \frac{1}{10}$	$-0.100335$	$-0.100000$	$0.000335$
$\frac{\pi}{2} + \frac{1}{2}$	$-0.546302$	$-0.500000$	$0.046302$
$\frac{\pi}{2} + 1$	$-1.557408$	$-1.000000$	$0.557408$

\cot x

\tilde f

There is no reference point at 0 — cot blows up there. Away from 0 it behaves like anything else.

2.1.6 $\csc x$

\csc x

\approx

\csc x_0 - \csc x_0 \cot x_0\,(x - x_0)

reference

\frac{\pi}{4}

\frac{\pi}{2}

\frac{\pi}{4}

\csc x

\approx

\csc\frac{\pi}{4} - \csc\frac{\pi}{4}\cot\frac{\pi}{4}\left(x - \frac{\pi}{4}\right)

\approx

\sqrt{2} - \sqrt{2}\left(x - \frac{\pi}{4}\right)

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{4} + 0$	$1.414214$	$1.414214$	$0.000000$
$\frac{\pi}{4} + \frac{1}{100}$	$1.400281$	$1.400071$	$0.000210$
$\frac{\pi}{4} + \frac{1}{10}$	$1.291711$	$1.272792$	$0.018919$
$\frac{\pi}{4} + \frac{1}{2}$	$1.042156$	$0.707107$	$0.335049$
$\frac{\pi}{4} + 1$	$1.023477$	$-2.22 \times 10^{-16}$	$1.023477$

\csc x

\tilde f

\frac{\pi}{2}

\csc x

\approx

\csc\frac{\pi}{2} - \csc\frac{\pi}{2}\cot\frac{\pi}{2}\left(x - \frac{\pi}{2}\right)

\approx

1

	$f(x)$	$\tilde f(x)$	error
$\frac{\pi}{2} + 0$	$1.000000$	$1.000000$	$0.000000$
$\frac{\pi}{2} + \frac{1}{100}$	$1.000050$	$1.000000$	$5.00 \times 10^{-5}$
$\frac{\pi}{2} + \frac{1}{10}$	$1.005021$	$1.000000$	$0.005021$
$\frac{\pi}{2} + \frac{1}{2}$	$1.139494$	$1.000000$	$0.139494$
$\frac{\pi}{2} + 1$	$1.850816$	$1.000000$	$0.850816$

\csc x

\tilde f

Same as cot — undefined at 0, fine everywhere else. At π/2 it bottoms out, so the line is flat.

2.2 Exponents and logarithms

2.2.1 $e^x$

e^x

\approx

e^{x_0} + e^{x_0}\,(x - x_0)

reference

0

1

2

0

e^x

\approx

e^0 + e^0\,(x - 0)

\approx

1 + x

	$f(x)$	$\tilde f(x)$	error
$0 + 0$	$1.000000$	$1.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.010050$	$1.010000$	$5.02 \times 10^{-5}$
$0 + \frac{1}{10}$	$1.105171$	$1.100000$	$0.005171$
$0 + \frac{1}{2}$	$1.648721$	$1.500000$	$0.148721$
$0 + 1$	$2.718282$	$2.000000$	$0.718282$

e^x

\tilde f

1

e^x

\approx

e^1 + e^1\,(x - 1)

\approx

e\,x

	$f(x)$	$\tilde f(x)$	error
$1 + 0$	$2.718282$	$2.718282$	$0.000000$
$1 + \frac{1}{100}$	$2.745601$	$2.745465$	$0.000136$
$1 + \frac{1}{10}$	$3.004166$	$2.990110$	$0.014056$
$1 + \frac{1}{2}$	$4.481689$	$4.077423$	$0.404266$
$1 + 1$	$7.389056$	$5.436564$	$1.952492$

e^x

\tilde f

2

e^x

\approx

e^2 + e^2\,(x - 2)

\approx

e^2\left(x - 1\right)

	$f(x)$	$\tilde f(x)$	error
$2 + 0$	$7.389056$	$7.389056$	$0.000000$
$2 + \frac{1}{100}$	$7.463317$	$7.462947$	$0.000371$
$2 + \frac{1}{10}$	$8.166170$	$8.127962$	$0.038208$
$2 + \frac{1}{2}$	$12.182494$	$11.083584$	$1.098910$
$2 + 1$	$20.085537$	$14.778112$	$5.307425$

e^x

\tilde f

At x₀ = 1 the two terms collapse: e + e(x − 1) = e·x.

2.2.2 $\ln(1+x)$

\ln(1+x)

\approx

\ln(1+x_0) + \dfrac{x - x_0}{1 + x_0}

reference

0

1

2

0

\ln(1+x)

\approx

\ln(1+0) + \dfrac{x - 0}{1 + 0}

\approx

x

	$f(x)$	$\tilde f(x)$	error
$0 + 0$	$0.000000$	$0.000000$	$0.000000$
$0 + \frac{1}{100}$	$0.009950$	$0.010000$	$4.97 \times 10^{-5}$
$0 + \frac{1}{10}$	$0.095310$	$0.100000$	$0.004690$
$0 + \frac{1}{2}$	$0.405465$	$0.500000$	$0.094535$
$0 + 1$	$0.693147$	$1.000000$	$0.306853$

\ln(1+x)

\tilde f

1

\ln(1+x)

\approx

\ln(1+1) + \dfrac{x - 1}{1 + 1}

\approx

\ln 2 + \frac{x - 1}{2}

	$f(x)$	$\tilde f(x)$	error
$1 + 0$	$0.693147$	$0.693147$	$0.000000$
$1 + \frac{1}{100}$	$0.698135$	$0.698147$	$1.25 \times 10^{-5}$
$1 + \frac{1}{10}$	$0.741937$	$0.743147$	$0.001210$
$1 + \frac{1}{2}$	$0.916291$	$0.943147$	$0.026856$
$1 + 1$	$1.098612$	$1.193147$	$0.094535$

\ln(1+x)

\tilde f

2

\ln(1+x)

\approx

\ln(1+2) + \dfrac{x - 2}{1 + 2}

\approx

\ln 3 + \frac{x - 2}{3}

	$f(x)$	$\tilde f(x)$	error
$2 + 0$	$1.098612$	$1.098612$	$0.000000$
$2 + \frac{1}{100}$	$1.101940$	$1.101946$	$5.54 \times 10^{-6}$
$2 + \frac{1}{10}$	$1.131402$	$1.131946$	$0.000544$
$2 + \frac{1}{2}$	$1.252763$	$1.265279$	$0.012516$
$2 + 1$	$1.386294$	$1.431946$	$0.045651$

\ln(1+x)

\tilde f

Written as ln(1 + x) so that 0 is a usable reference point. Plain ln x has none.

2.2.3 $(1+x)^r$

	$f(x)$	$\tilde f(x)$	error
$1 + 0$	$1.414214$	$1.414214$	$0.000000$
$1 + \frac{1}{100}$	$1.417745$	$1.417749$	$4.41 \times 10^{-6}$
$1 + \frac{1}{10}$	$1.449138$	$1.449569$	$0.000431$
$1 + \frac{1}{2}$	$1.581139$	$1.590990$	$0.009851$
$1 + 1$	$1.732051$	$1.767767$	$0.035716$

	$f(x)$	$\tilde f(x)$	error
$1 + 0$	$0.707107$	$0.707107$	$0.000000$
$1 + \frac{1}{100}$	$0.705346$	$0.705339$	$6.60 \times 10^{-6}$
$1 + \frac{1}{10}$	$0.690066$	$0.689429$	$0.000636$
$1 + \frac{1}{2}$	$0.632456$	$0.618718$	$0.013737$
$1 + 1$	$0.577350$	$0.530330$	$0.047020$

	$f(x)$	$\tilde f(x)$	error
$1 + 0$	$0.500000$	$0.500000$	$0.000000$
$1 + \frac{1}{100}$	$0.497512$	$0.497500$	$1.24 \times 10^{-5}$
$1 + \frac{1}{10}$	$0.476190$	$0.475000$	$0.001190$
$1 + \frac{1}{2}$	$0.400000$	$0.375000$	$0.025000$
$1 + 1$	$0.333333$	$0.250000$	$0.083333$

	$f(x)$	$\tilde f(x)$	error
$1 + 0$	$4.000000$	$4.000000$	$0.000000$
$1 + \frac{1}{100}$	$4.040100$	$4.040000$	$1.00 \times 10^{-4}$
$1 + \frac{1}{10}$	$4.410000$	$4.400000$	$0.010000$
$1 + \frac{1}{2}$	$6.250000$	$6.000000$	$0.250000$
$1 + 1$	$9.000000$	$8.000000$	$1.000000$

	$f(x)$	$\tilde f(x)$	error
$1 + 0$	$8.000000$	$8.000000$	$0.000000$
$1 + \frac{1}{100}$	$8.120601$	$8.120000$	$0.000601$
$1 + \frac{1}{10}$	$9.261000$	$9.200000$	$0.061000$
$1 + \frac{1}{2}$	$15.625000$	$14.000000$	$1.625000$
$1 + 1$	$27.000000$	$20.000000$	$7.000000$

The workhorse. r = 1/2 gives √(1+x), and r = −1 gives 1/(1+x).

2.2.4 $a^x$

a =2310

a^x

\approx

a^{x_0} + a^{x_0}\ln a\,(x - x_0)

reference

0

1

0

a^x

\approx

a^0 + a^0\ln a\,(x - 0)

\approx

1 + x\ln a

	$f(x)$	$\tilde f(x)$	error
$0 + 0$	$1.000000$	$1.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.006956$	$1.006931$	$2.41 \times 10^{-5}$
$0 + \frac{1}{10}$	$1.071773$	$1.069315$	$0.002459$
$0 + \frac{1}{2}$	$1.414214$	$1.346574$	$0.067640$
$0 + 1$	$2.000000$	$1.693147$	$0.306853$

a^x

\tilde f

1

a^x

\approx

a^1 + a^1\ln a\,(x - 1)

\approx

a + a\ln a\,(x - 1)

	$f(x)$	$\tilde f(x)$	error
$1 + 0$	$2.000000$	$2.000000$	$0.000000$
$1 + \frac{1}{100}$	$2.013911$	$2.013863$	$4.82 \times 10^{-5}$
$1 + \frac{1}{10}$	$2.143547$	$2.138629$	$0.004917$
$1 + \frac{1}{2}$	$2.828427$	$2.693147$	$0.135280$
$1 + 1$	$4.000000$	$3.386294$	$0.613706$

a^x

\tilde f

a^x

\approx

a^{x_0} + a^{x_0}\ln a\,(x - x_0)

reference

0

1

0

a^x

\approx

a^0 + a^0\ln a\,(x - 0)

\approx

1 + x\ln a

	$f(x)$	$\tilde f(x)$	error
$0 + 0$	$1.000000$	$1.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.011047$	$1.010986$	$6.06 \times 10^{-5}$
$0 + \frac{1}{10}$	$1.116123$	$1.109861$	$0.006262$
$0 + \frac{1}{2}$	$1.732051$	$1.549306$	$0.182745$
$0 + 1$	$3.000000$	$2.098612$	$0.901388$

a^x

\tilde f

1

a^x

\approx

a^1 + a^1\ln a\,(x - 1)

\approx

a + a\ln a\,(x - 1)

	$f(x)$	$\tilde f(x)$	error
$1 + 0$	$3.000000$	$3.000000$	$0.000000$
$1 + \frac{1}{100}$	$3.033140$	$3.032958$	$0.000182$
$1 + \frac{1}{10}$	$3.348370$	$3.329584$	$0.018786$
$1 + \frac{1}{2}$	$5.196152$	$4.647918$	$0.548234$
$1 + 1$	$9.000000$	$6.295837$	$2.704163$

a^x

\tilde f

a^x

\approx

a^{x_0} + a^{x_0}\ln a\,(x - x_0)

reference

0

1

0

a^x

\approx

a^0 + a^0\ln a\,(x - 0)

\approx

1 + x\ln a

	$f(x)$	$\tilde f(x)$	error
$0 + 0$	$1.000000$	$1.000000$	$0.000000$
$0 + \frac{1}{100}$	$1.023293$	$1.023026$	$0.000267$
$0 + \frac{1}{10}$	$1.258925$	$1.230259$	$0.028667$
$0 + \frac{1}{2}$	$3.162278$	$2.151293$	$1.010985$
$0 + 1$	$10.000000$	$3.302585$	$6.697415$

a^x

\tilde f

1

a^x

\approx

a^1 + a^1\ln a\,(x - 1)

\approx

a + a\ln a\,(x - 1)

	$f(x)$	$\tilde f(x)$	error
$1 + 0$	$10.000000$	$10.000000$	$0.000000$
$1 + \frac{1}{100}$	$10.232930$	$10.230259$	$0.002671$
$1 + \frac{1}{10}$	$12.589254$	$12.302585$	$0.286669$
$1 + \frac{1}{2}$	$31.622777$	$21.512925$	$10.109851$
$1 + 1$	$100.000000$	$33.025851$	$66.974149$

a^x

\tilde f

Covers e^x too. Set a = e, so ln a = 1.

3. Algebra

You do not have to differentiate a messy function at all. You can build its approximation out of the entries above instead.

Write $L(f)$ for the linear approximation of $f$ near $0$ .

The useful fact is that you can approximate first and combine second. For a product:

L\big(L(f)\,L(g)\big) = L(fg)

So the $L$ of a product is just the linear approximation of the product of the linear approximations. You never have to touch $fg$ itself.

Here is the full set:

	rule
sum	$L(f+g) = L(f) + L(g)$
scale	$L(kf) = k\,L(f)$
product	$L\big(L(f)\,L(g)\big) = L(fg)$
composition, when $g(0)=0$	$L\big(L(f)\circ L(g)\big) = L(f\circ g)$

Applied to a polynomial, $L$ does nothing but chop off the high powers. So the outer $L$ on each left-hand side is the cut — the $x^2$ term a product or composition produces along the way is not trustworthy, since the inputs were only accurate to first order.

The $g(0) = 0$ condition on the last line is not a deep restriction — it falls out of how $L$ is defined here. $L(f)$ only carries information about $f$ near $0$ , so plugging in $g(x)$ is only valid where $g$ actually lands near $0$ . Centre $L$ at a different reference point and the condition moves with it: substitution is valid whenever the inside function's value at your reference point equals the outside function's reference point.

1. Definition

2. Cheatsheet

2.1 Trigonometric

2.1.1sin⁡x\sin xsinx

2.1.2cos⁡x\cos xcosx

2.1.3tan⁡x\tan xtanx

2.1.4sec⁡x\sec xsecx

2.1.5cot⁡x\cot xcotx

2.1.6csc⁡x\csc xcscx

2.2 Exponents and logarithms

2.2.1exe^xex

2.2.2ln⁡(1+x)\ln(1+x)ln(1+x)

2.2.3(1+x)r(1+x)^r(1+x)r

2.2.4axa^xax

3. Algebra

2.1.1 $\sin x$

2.1.2 $\cos x$

2.1.3 $\tan x$

2.1.4 $\sec x$

2.1.5 $\cot x$

2.1.6 $\csc x$

2.2.1 $e^x$

2.2.2 $\ln(1+x)$

2.2.3 $(1+x)^r$

2.2.4 $a^x$