Notation for the Second Derivative
Key Questions
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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
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Relationship between First and Second Derivatives of a Function
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Analyzing Concavity of a Function
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Notation for the Second Derivative
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Determining Points of Inflection for a Function
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First Derivative Test vs Second Derivative Test for Local Extrema
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The special case of x⁴
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Critical Points of Inflection
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Application of the Second Derivative (Acceleration)
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Examples of Curve Sketching