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Monday, 17 July 2017

Derivatives and Integration of absolute value functions


The proofs below will use the absolute value functions form as

|f(x)|=\sqrt{f(x)^2}

Claim 1:  {d\over dx}|x|={x\over |x|}
Proof:

Claim 2:  {d\over dx}|f(x)|={f(x)\over |f(x)|}f'(x)

Proof:

Claim 3: \int |x| dx={x|x|\over 2}+c
Proof:
Claim 4: \int x|x| dx={x^2|x|\over 3}+c
Proof:

Yes it does generalise,

Claim 5: \int x^n|x| dx={x^{n+1}|x|\over n+2}+c
Proof:

Claim 6: \int |f(x)| dx=??




Proof: