# Chain rule

#### Everything You Need in One Place

Homework problems? Exam preparation? Trying to grasp a concept or just brushing up the basics? Our extensive help & practice library have got you covered.

#### Learn and Practice With Ease

Our proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals.

#### Instant and Unlimited Help

Our personalized learning platform enables you to instantly find the exact walkthrough to your specific type of question. Activate unlimited help now!

0/1
##### Intros
###### 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. $\frac{d}{dx} \log_{2}{x^3}$
###### 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