# Definition of derivative

##### Examples
###### Lessons
1. Find the derivative of the given function using the definition of derivative.
$f\left( x \right) = {x^3} - 5x + 6$
1. Definition of derivative with irregular functions
Find the derivative of the following functions using the definition of derivative.
1. $f(x)=\sqrt{x-2}$
2. $f(x)=\frac{3-x}{2+x}$
2. Applications to definition of derivative
Let $f(x)=4x^{\frac{1}{3}}$
1. For when $x \,$$\, 0$, find the derivative of $f(x)$.
2. Show that $f'(0)$ does not exist.
3. For what value(s) of $x$ does the vertical tangent line occur?
###### 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
We have studied the notion of average rate of change thus far, for example, change in position over time (velocity), average change in velocity over time (acceleration) etc. However, what if we are interested in finding the instantaneous rate of change of something? To answer this, we will first learn about the concept of the definition of derivative in this section, as well as how to apply it.
Definition of Derivative
$f'\left( x \right) = \;_{h \to 0}^{\;lim}\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}$