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
ANSWERED
asked 2021-06-01
Use the theorems on derivatives to find the derivatives of the following function:
\(\displaystyle{f{{\left({x}\right)}}}={\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}\)

Answers (1)

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
\(\displaystyle{f{{\left({x}\right)}}}={\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}\)
\(\displaystyle{f}'{\left({x}\right)}={\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}+{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}\)
\(\displaystyle{f}'{\left({x}\right)}={\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}{\left[{32}{x}^{{{3}}}-{6}{x}^{{{2}}}\right]}+{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}{\left[{4}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{3}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}\right]}\)
\(\displaystyle{f}'{\left({x}\right)}={2}{x}^{{{2}}}{\left({16}{x}-{3}\right)}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}+{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}{\left[{4}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{3}}}{\left({15}{x}^{{{2}}}+{16}{x}\right)}\right]}\)
\(\displaystyle{f}'{\left({x}\right)}={2}{x}^{{{2}}}{\left({16}{x}-{3}\right)}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{4}}}+{4}{x}{\left({15}{x}+{16}\right)}{\left({8}{x}^{{{4}}}-{2}{x}^{{{3}}}-{7}\right)}{\left({5}{x}^{{{3}}}+{8}{x}^{{{2}}}-{4}\right)}^{{{3}}}\)
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