Notation for the Second Derivative
Key Questions

If you prefer Leibniz notation, second derivative is denoted
#(d^2y)/(dx^2)# .Example:
#y = x^2# #dy/dx = 2x# #(d^2y)/(dx^2) = 2# If you like the primes notation, then second derivative is denoted with two prime marks, as opposed to the one mark with first derivatives:
#y = x^2#
#y' = 2x#
#y'' = 2# Similarly, if the function is in function notation:
#f(x) = x^2#
#f'(x) = 2x#
#f''(x) = 2# Most people are familiar with both notations, so it doesn't usually matter which notation you choose, so long as people can understand what you're writing. I myself prefer the Leibniz notation, because otherwise I tend to confuse the apostrophes with exponents of one or eleven. Though the primes notation is more shorthand and quicker to write, so many people prefer it.
Questions
Graphing with the Second Derivative

Relationship between First and Second Derivatives of a Function

Analyzing Concavity of a Function

Notation for the Second Derivative

Determining Points of Inflection for a Function

First Derivative Test vs Second Derivative Test for Local Extrema

The special case of x⁴

Critical Points of Inflection

Application of the Second Derivative (Acceleration)

Examples of Curve Sketching