Calculus l - Differentiation

What is a Derivative? 

The derivative is a function that is derived from the original function using the process of differentiation. It is used to find the slope at some x-value. When the derivative is graphed with respect to its x-value, it can be used to analyze the slope of a line tangent to the point that exists on the original function’s graph. Whenever the derivative is negative, we know that the slope of the tangent line during that interval [x1, x2] is also negative (going downwards) and vice versa.  

Differentiation Notations: 

 

Definition of the derivative: 

 

Basic rules: 

Constant multiplied to a function rule: 

 

 

For single variable polynomials:  
 

Adding two functions: 

 

Subtracting two functions: 

 

 

Helpful-to-have-memorized: 

         

Trig derivatives: 

 

 

Formula for derivatives of natural logs: 

 

Formula for derivatives of logarithms: 

 

Formula for derivatives of exponentials: 

 

 

 

 

 
 

***Remember: a function is continuous if it is differentiable, but not necessarily differentiable if it is continuous.*** 

Jump discontinuities are always non-differentiable, but they may have one-sided derivatives. Wherever there is a vertical line, it’ll be non-differentiable. Sharp turns are undifferentiable because the left and right side derivatives don’t match, so we don’t know which way the tangent line would be angled at that point.  

 

 

 
 

 

 
 

Here’s a modified one you can use when you have a function inside a function inside a function... 

 

 

Implicit differentiation: used when x and y are not related in a simple manner like    x = y    or   y = f(x). 

 
 

However, here is an example when you would not really need to use the quotient rule: 
 

 

 

 

 
 
Some tougher examples I did for fun 😊: