The P3 derivative set

8 minA Level3.4.1

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can differentiate $e^{x}$ , $\ln x$ , $\sin x$ , $\cos x$ , $\tan x$ and $\tan^{-1}x$ , including versions with a bracket inside like $e^{3x}$ or $\sin(2x + 1)$ .

Key to the symbols

Symbol	Meaning
$\dfrac{d}{dx}$	"the derivative of", with respect to $x$
$e^x$	the exponential function ( $e \approx 2.718$ )
$\ln x$	the natural logarithm, $\log_e x$
$\tan^{-1}x$	the inverse tan function: the angle whose tan is $x$ (not $\dfrac{1}{\tan x}$ )
$y'$ or $f'(x)$	another way to write $\dfrac{dy}{dx}$ (read "y dash")
$\sec x$	$\dfrac{1}{\cos x}$ , so $\sec^2 x = \dfrac{1}{\cos^2 x}$

Angles are in radians. These derivatives are only true when $x$ is measured in radians.

Definition

The P3 standard derivatives

$y$	$\dfrac{dy}{dx}$
$e^x$	$e^x$
$\ln x$	$\dfrac{1}{x}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	$\sec^2 x$
$\tan^{-1}x$	$\dfrac{1}{1 + x^2}$

Watch the signs: only $\cos x$ picks up a minus.

With a bracket inside - the chain rule

When the function has something other than plain $x$ inside, use the chain rule from P1:

Differentiate the outside function, leaving the inside exactly the same.
Multiply by the derivative of the inside.

Examples:

$e^{3x}$ : outside $e^{(\ )}$ , inside $3x$ . Derivative $e^{3x} \times 3 = 3e^{3x}$ .
$\sin(2x + 1)$ : outside $\sin$ , inside $2x + 1$ . Derivative $\cos(2x + 1) \times 2$ .
$\ln(x^2 + 1)$ : outside $\ln$ , inside $x^2 + 1$ . Derivative $\dfrac{1}{x^2 + 1} \times 2x = \dfrac{2x}{x^2 + 1}$ .

A shortcut worth knowing: if the inside is $kx$ , the derivative just gains a factor $k$ . So $\dfrac{d}{dx}\sin kx = k\cos kx$ and $\dfrac{d}{dx}e^{kx} = ke^{kx}$ .

Simplify first when it helps

Sometimes it's quicker to rewrite before differentiating.

Log laws can split a log: $\ln(5x) = \ln 5 + \ln x$ . Since $\ln 5$ is a constant, its derivative is 0.
Expanding can remove a bracket: $(e^x + 1)^2 = e^{2x} + 2e^x + 1$ .
Recognising a function: $\dfrac{\sin x}{\cos x} = \tan x$ .

Both routes give the same answer. Choose the one with fewer steps.

Worked example: Example 1: two of the new ones

Differentiate $y = e^{-x^2} + \ln(5x)$ .

Have a go first?

Quick check

Multiple choiceQuestion 1 · Recall

Answer the 3 questions above to continue.