Chain rule

Chain Rule appears everywhere in the world of differential calculus. Whenever we are finding the derivative of a function, be it a composite function or not, we are in fact using the Chain Rule. In this section, we will learn about the concept, the definition and the application of the Chain Rule, as well as a secret trick – "The Bracket Technique".

Lessons

Notes:

Chain Rule
if: $y = \;f\left( {\;\;\;\;\;\;\;} \right)$
then: $\frac{{dy}}{{{d}x}} = f'\left( {\;\;\;\;\;\;\;} \right)\cdot\frac{{d}}{{{d}x}}\left( {\;\;\;\;\;\;\;\;} \right)$

Differential Rules

3.
Differentiate: Rational Functions
4.
Differentiate: Radical Functions
5.
Differentiate: Trigonometric Functions
6.
Differentiate: Exponential Functions
7.
Differentiate: Logarithmic Functions

Chain rule

Don't just watch, practice makes perfect.

We have over 170 practice questions in Calculus 1 for you to master.

Get Started Now

Let's keep going.

Play next lesson

Practice this topic

Let's keep going.

Play next lesson

Let's learn the next topic

Goto next topic

Let's keep going

Let's keep going!

Play next lesson or Practice this topic

Start now and get better math marks!

Keep exploring StudyPug

Start now and get better math marks!

Keep exploring StudyPug

Lesson: 1

Selected

Lesson: 2

Lesson: 3a

Lesson: 3b

Lesson: 4a

Lesson: 4b

Lesson: 4c

Lesson: 4d

Lesson: 4e

Lesson: 5a

Lesson: 5b

Lesson: 5c

Lesson: 6a

Lesson: 6b

Lesson: 6c

Lesson: 6d

Lesson: 7a

Lesson: 7b

Intro

Learn

Practice

Do better in math today

Don't just watch, practice makes perfect.

Do better in math today

Don't just watch, practice makes perfect.

Lessons

Notes:

Chain Rule if: y=f()y = \;f\left( {\;\;\;\;\;\;\;} \right)y=f() then: dydx=f′()⋅ddx()\frac{{dy}}{{{d}x}} = f'\left( {\;\;\;\;\;\;\;} \right)\cdot\frac{{d}}{{{d}x}}\left( {\;\;\;\;\;\;\;\;} \right)​dx​​dy​​=f​′​​()⋅​dx​​d​​() Differential Rules

1.

Introduction to Chain Rule • “bracket technique” explained! • exercise: ddxx10\frac{d}{dx}x^{10}​dx​​d​​x​10​​ VS. ddx(x5+4x3−6x+8)10\frac{d}{dx}(x^5+4x^3-6x+8)^{10}​dx​​d​​(x​5​​+4x​3​​−6x+8)​10​​

2.

Differentiate: Polynomial Functions ddx(2x−1)3 \frac{d}{dx} (2x-1)^3​dx​​d​​(2x−1)​3​​

3.

Differentiate: Rational Functions

a)

ddx1(4x3+7)10 \frac{d}{dx} \frac{1}{(4x^3+7)^{10}} ​dx​​d​​​(4x​3​​+7)​10​​​​1​​

b)

ddx−5sin2x \frac{d}{dx}- \frac{5}{\sin ^2x} ​dx​​d​​−​sin​2​​x​​5​​

4.

Differentiate: Radical Functions

a)

ddxx3+4x2−9\frac{d}{dx} \sqrt{x^3+4x^2-9} ​dx​​d​​√​x​3​​+4x​2​​−9​​​

b)

ddx3(x2+5)7 \frac{d}{dx} {^3}\sqrt{(x^2+5)^7} ​dx​​d​​​3​​√​(x​2​​+5)​7​​​​​

c)

ddx136x4−x\frac{d}{dx} \frac{1}{{^3}\sqrt{6x^4-x}} ​dx​​d​​​​3​​√​6x​4​​−x​​​​​1​​

d)

ddxx+x+x \frac{d}{dx} \sqrt{x+\sqrt{x+\sqrt{x}}} ​dx​​d​​√​x+√​x+√​x​​​​​​​​​

e)

ddx3lnx \frac{d}{dx} {^3}\sqrt{\ln x} ​dx​​d​​​3​​√​lnx​​​

5.

Differentiate: Trigonometric Functions

a)

Differentiate: y=sin4xy= \sin ^4x y=sin​4​​x VS. y=sin(x4)y=\sin (x^4)y=sin(x​4​​)

b)

ddxtan(cose5x2) \frac{d}{dx} \tan (\cos e^{5x^2}) ​dx​​d​​tan(cose​5x​2​​​​)

c)

ddθsin(cos(tanθ))\frac{d}{d \theta} \sin (\cos (\tan \theta)) ​dθ​​d​​sin(cos(tanθ))

6.

Differentiate: Exponential Functions

a)

ddxetanx\frac{d}{dx} e^{\tan x} ​dx​​d​​e​tanx​​

b)

ddxecsc5x2\frac{d}{dx} e^{\csc 5x^2} ​dx​​d​​e​csc5x​2​​​​

c)

or

Chain Rule
if: $y = \;f\left( {\;\;\;\;\;\;\;} \right)$
then: $\frac{{dy}}{{{d}x}} = f'\left( {\;\;\;\;\;\;\;} \right)\cdot\frac{{d}}{{{d}x}}\left( {\;\;\;\;\;\;\;\;} \right)$

Differential Rules

Introduction to Chain Rule
• “bracket technique” explained!
• exercise: $\frac{d}{dx}x^{10}$ VS. $\frac{d}{dx}(x^5+4x^3-6x+8)^{10}$

Differentiate: Polynomial Functions
$\frac{d}{dx} (2x-1)^3$

$\frac{d}{dx} \frac{1}{(4x^3+7)^{10}}$

$\frac{d}{dx}- \frac{5}{\sin ^2x}$

$\frac{d}{dx} \sqrt{x^3+4x^2-9}$

$\frac{d}{dx} {^3}\sqrt{(x^2+5)^7}$

$\frac{d}{dx} \frac{1}{{^3}\sqrt{6x^4-x}}$

$\frac{d}{dx} \sqrt{x+\sqrt{x+\sqrt{x}}}$

$\frac{d}{dx} {^3}\sqrt{\ln x}$

Differentiate: $y= \sin ^4x$
VS.
$y=\sin (x^4)$

$\frac{d}{dx} \tan (\cos e^{5x^2})$

$\frac{d}{d \theta} \sin (\cos (\tan \theta))$

$\frac{d}{dx} e^{\tan x}$

$\frac{d}{dx} e^{\csc 5x^2}$

$\frac{d}{dx} 2^{\sin x}$

$\frac{d}{dx} 5^{2^{{x}^3}}$

$\frac{d}{dx} \ln x^{100}$
VS.
$\frac{d}{dx} (\ln x)^{100}$

$\frac{d}{dx} \log_{2}{x^3}$