Question

Find the derivatives f(x)=(x^{2}-2x+1)(x^{2}-1)

Derivatives
ANSWERED
asked 2021-03-26
Find the derivatives
\(\displaystyle{f{{\left({x}\right)}}}={\left({x}^{{{2}}}-{2}{x}+{1}\right)}{\left({x}^{{{2}}}-{1}\right)}\)

Answers (1)

2021-03-28
Step 1
The given function is \(\displaystyle{f{{\left({x}\right)}}}={\left({x}^{{{2}}}-{2}{x}+{1}\right)}{\left({x}^{{{2}}}-{1}\right)}\)
Step 2
Using formula
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({f}.{g}\right)}={g}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{f}+{f}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{g}\)
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{x}^{{{n}}}={n}{x}^{{{n}-{1}}}\)
Step 3
On differentiating
\(\displaystyle{f{{\left({x}\right)}}}={\left({x}^{{{2}}}-{2}{x}+{1}\right)}{\left({x}^{{{2}}}-{1}\right)}\)
\(\displaystyle{\frac{{{d}{f{{\left({x}\right)}}}}}{{{\left.{d}{x}\right.}}}}={\left({x}^{{{2}}}-{1}\right)}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{2}}}-{2}{x}+{1}\right)}+{\left({x}^{{{2}}}-{2}{x}+{1}\right)}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{2}}}-{1}\right)}\)
\(\displaystyle{f}'{\left({x}\right)}={\left({x}^{{{2}}}-{1}\right)}{\left({2}{x}-{2}\right)}+{\left({x}^{{{2}}}-{2}{x}+{1}\right)}{\left({2}{x}\right)}\)
\(\displaystyle{f}'{\left({x}\right)}={\left({2}{x}^{{{3}}}-{2}{x}^{{{2}}}-{2}{x}+{2}\right)}+{\left({2}{x}^{{{3}}}-{4}{x}^{{{2}}}+{2}{x}\right)}\)
\(\displaystyle{f}'{\left({x}\right)}={4}{x}^{{{3}}}-{6}{x}^{{{2}}}+{2}\)
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