Question

# Use the theorems on derivatives to find the derivatives of the following function: f(x)=(5x^{3}+8x^{2}-4)^{4}(8x^{4}-2x^{3}-7)

Derivatives
Use the theorems on derivatives to find the derivatives of the following function:
$$\displaystyle{f{{\left({x}\right)}}}={\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}$$

2021-06-02
Step 1
Given
$$\displaystyle{f{{\left({x}\right)}}}={\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}$$
Step 2
Find the derivative
$$\displaystyle{f{{\left({x}\right)}}}={\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}$$
$$\displaystyle{f}'{\left({x}\right)}={\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}+{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}$$
$$\displaystyle{f}'{\left({x}\right)}={\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}{\left[{32}{x}^{{{3}}}-{6}{x}^{{{2}}}\right]}+{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}{\left[{4}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{3}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}\right]}$$
$$\displaystyle{f}'{\left({x}\right)}={2}{x}^{{{2}}}{\left({16}{x}-{3}\right)}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}+{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}{\left[{4}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{3}}}{\left({15}{x}^{{{2}}}+{16}{x}\right)}\right]}$$
$$\displaystyle{f}'{\left({x}\right)}={2}{x}^{{{2}}}{\left({16}{x}-{3}\right)}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}+{4}{x}{\left({15}{x}+{16}\right)}{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{3}}}$$