Name 
Notation 
First Derivative 
, , , , 
Second Derivative 
, , , , 
Third Derivative 
, , ,, 
n^{th }Derivative 
, , , , 


When more than one derivative
is applied to a function, it is consider higher order derivatives. For
example, the derivative of y = x^{7} is
7x^{6}.
7x^{6} is differentiable and
its derivative is 42x^{5}. 42x^{5} is called the second
order derivative of y with respect to x for function x^{7}. Repeating
this process, function y's third order derivative
210x^{4 }is obtained. The higher order derivative
notations is shown on the left.
The chain rule also holds for higher order derivatives.
For instance, one can find the fourth derivative for
y = A sin(ωt
+ 1), in which y is a function of t.
Consider y is a function of θ where θ = ωt
+1. Therefore, the first order derivative is
The second order derivative is
The third order derivative is
The forth order derivative is
The implicit differentiation method also holds for higher
order derivatives. For example, one can find the second derivative for
ellipse
.
In ellipse, y is a function of x. In order to find the
derivative of y with respect to x, implicit differentiation method is
applied. The first derivative is
(1)
Rearranging equation (1) gives
(2)
Taking another derivative with respect to x gives
(3)
Substitute equation (2) to (3),
(4) 