a) Find the derivative of f(x)=(2x+9)(8x−7) by first expanding the polynomials.

postillan4 2021-09-16 Answered

a) Find the derivative of \(f(x)=(2x+9)(8x−7)\) by first expanding the polynomials.
b) Find the derivative of \(f(x)=(2x+9)(8x−7)\) by using the product rule. Let \(\displaystyle g{{\left({x}\right)}}={2}{x}+{9}{\quad\text{and}\quad}{h}{\left({x}\right)}={8}{x}-{7}\)

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Expert Answer

Leonard Stokes
Answered 2021-09-17 Author has 7567 answers

a) \(f(x)=(2x+9)(8x-7)\)
\(\displaystyle{f{{\left({x}\right)}}}={16}{x}^{{2}}-{14}{x}+{72}{x}-{63}\)
\(\displaystyle{f{{\left({x}\right)}}}={16}{x}^{{2}}+{58}{x}-{63}\)
\(\displaystyle{f}'{\left({x}\right)}={32}{x}+{58}\)
b) \(\displaystyle{f{{\left({x}\right)}}}={\left({2}{x}+{9}\right)}{\left({8}{x}-{7}\right)}\)
\(\displaystyle{g{{\left({x}\right)}}}={\left({2}{x}+{9}\right)}\)
\(\displaystyle{h}{\left({x}\right)}={8}{x}-{7}\)
\(\displaystyle{g}'{\left({x}\right)}={2}\)
\(\displaystyle{h}'{\left({x}\right)}={8}\)
\(\displaystyle{f{{\left({x}\right)}}}={\left({2}{x}+{9}\right)}{\left({8}{x}-{7}\right)}={g{{\left({x}\right)}}}\cdot{h}{\left({x}\right)}\)
\(\displaystyle{f}'{\left({x}\right)}={g{{\left({x}\right)}}}\cdot{h}'{\left({x}\right)}+{h}{\left({x}\right)}\cdot{g}'{\left({x}\right)}\)
\(\displaystyle{f}'{\left({x}\right)}={\left({2}{x}+{9}\right)}{\left({8}\right)}+{\left({8}{x}-{7}\right)}{\left({2}\right)}\)
\(\displaystyle{f}'{\left({x}\right)}={16}{x}+{72}+{16}{x}-{14}\)
\(\displaystyle{f}'{\left({x}\right)}={32}{x}-{58}\)

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