# Chain rule

0/1
##### Introduction
###### Lessons
1. Introduction to Chain Rule
• "bracket technique" explained!
exercise: $\frac{d}{dx}x^{10}$ VS. $\frac{d}{dx}(x^5+4x^3-6x+8)^{10}$
0/17
##### Examples
###### Lessons
1. Differentiate: Polynomial Functions
$\frac{d}{dx} (2x-1)^3$
1. Differentiate: Rational Functions
1. $\frac{d}{dx} \frac{1}{(4x^3+7)^{10}}$
2. $\frac{d}{dx}- \frac{5}{\sin ^2x}$
1. $\frac{d}{dx} \sqrt{x^3+4x^2-9}$
2. $\frac{d}{dx} {^3}\sqrt{(x^2+5)^7}$
3. $\frac{d}{dx} \frac{1}{{^3}\sqrt{6x^4-x}}$
4. $\frac{d}{dx} \sqrt{x+\sqrt{x+\sqrt{x}}}$
5. $\frac{d}{dx} {^3}\sqrt{\ln x}$
3. Differentiate: Trigonometric Functions
1. Differentiate: $y= \sin ^4x$
VS.
$y=\sin (x^4)$
2. $\frac{d}{dx} \tan (\cos e^{5x^2})$
3. $\frac{d}{d \theta} \sin (\cos (\tan \theta))$
4. Differentiate: Exponential Functions
1. $\frac{d}{dx} e^{\tan x}$
2. $\frac{d}{dx} e^{\csc 5x^2}$
3. $\frac{d}{dx} 2^{\sin x}$
4. $\frac{d}{dx} 5^{2^{{x}^3}}$
5. Differentiate: Logarithmic Functions
1. $\frac{d}{dx} \ln x^{100}$
VS.
$\frac{d}{dx} (\ln x)^{100}$
2. $\frac{d}{dx} \log_{2}{x^3}$
###### Free to Join!
StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun - with achievements, customizable avatars, and awards to keep you motivated.
• #### Easily See Your Progress

We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.
• #### Make Use of Our Learning Aids

###### Practice Accuracy

See how well your practice sessions are going over time.

Stay on track with our daily recommendations.

• #### Earn Achievements as You Learn

Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.
• #### Create and Customize Your Avatar

Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.
###### Topic Notes
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".
Chain Rule
if: $y = \;f\left( {\;\;\;\;\;\;\;} \right)$
then: $\frac{{dy}}{{{d}x}} = f'\left( {\;\;\;\;\;\;\;} \right)\cdot\frac{{d}}{{{d}x}}\left( {\;\;\;\;\;\;\;\;} \right)$

Differential Rules