Continuity and Differentiability — Part 2: Derivatives and the chain rule: visual lesson, ISC Class 12 Mathematics

Lesson

Continuity and Differentiability, Part 2

A short table of standard derivatives and one rule for layers let you differentiate almost anything in this chapter.

In this lesson you will

Differentiate trigonometric, exponential and logarithmic functions
Apply the chain rule to composites with two or three layers
Differentiate a to the power x and logarithms of functions
Differentiate the inverse trigonometric functions
Simplify by trigonometric substitution before differentiating, checking the principal range

Part 2 at a glance.

Concept

Every chain-rule answer is built from this table.

Learn these by heart; the chain rule does the rest.
$f(x)$	$f'(x)$	$f(x)$	$f'(x)$
$\sin x$	$\cos x$	$e^x$	$e^x$
$\cos x$	$-\sin x$	$a^x$	$a^x\log a$
$\tan x$	$\sec^2 x$	$\log x$	$\dfrac{1}{x}$
$\sec x$	$\sec x\tan x$	$\operatorname{cosec} x$	$-\operatorname{cosec} x\cot x$

Diagram

At $(0, 1)$ the slope of $e^x$ is $1$; at $(1, 0)$ the slope of $\log x$ is $1$.

$\dfrac{d}{dx}e^x = e^x$ and $\dfrac{d}{dx}\log x = \dfrac{1}{x}$ .