Question

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

Derivatives
Find the derivatives
$$\displaystyle{f{{\left({x}\right)}}}={\left({x}^{{{2}}}-{2}{x}+{1}\right)}{\left({x}^{{{2}}}-{1}\right)}$$

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
$$\displaystyle{f{{\left({x}\right)}}}={\left({x}^{{{2}}}-{2}{x}+{1}\right)}{\left({x}^{{{2}}}-{1}\right)}$$
$$\displaystyle{\frac{{{d}{f{{\left({x}\right)}}}}}{{{\left.{d}{x}\right.}}}}={\left({x}^{{{2}}}-{1}\right)}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{2}}}-{2}{x}+{1}\right)}+{\left({x}^{{{2}}}-{2}{x}+{1}\right)}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}^{{{2}}}-{1}\right)}$$
$$\displaystyle{f}'{\left({x}\right)}={\left({x}^{{{2}}}-{1}\right)}{\left({2}{x}-{2}\right)}+{\left({x}^{{{2}}}-{2}{x}+{1}\right)}{\left({2}{x}\right)}$$
$$\displaystyle{f}'{\left({x}\right)}={\left({2}{x}^{{{3}}}-{2}{x}^{{{2}}}-{2}{x}+{2}\right)}+{\left({2}{x}^{{{3}}}-{4}{x}^{{{2}}}+{2}{x}\right)}$$
$$\displaystyle{f}'{\left({x}\right)}={4}{x}^{{{3}}}-{6}{x}^{{{2}}}+{2}$$