What is the first derivative and second derivative of #x^4 - 1#?

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Answer 1 · 2016-05-31T05:08:12

#f^'(x)=4x^3#
#f^''(x)=12x^2#

to find the first derivative we must simply use three rules:

1. Power rule
#d/dx x^n = nx^(n-1)#

2. Constant rule
#d/dx (c) = 0# (where c is an integer and not a variable)

3. Sum and difference rule
#d/dx [f(x)+-g(x)] = [f^'(x)+-g^'(x)]#

the first derivative results in:
#4x^3-0#
which simplifies to
#4x^3#

to find the second derivative, we must derive the first derivative by again applying the power rule which results in:
#12x^3#

you can keep going if you like:
third derivative = #36x^2#
fourth derivative = #72x#
fifth derivative = #72#
sixth derivative = #0#