Question

Calculate the derivatives of all orders: f'(x),f''(x),f'''(x),f^{4}(x),...f^{n}(x),...f(x)=e^{2x}

Derivatives
ANSWERED
asked 2021-06-27
Calculate the derivatives of all orders:
\(\displaystyle{f}'{\left({x}\right)},{f}{''}{\left({x}\right)},{f}{'''}{\left({x}\right)},{{f}^{{{4}}}{\left({x}\right)}},\ldots{{f}^{{{n}}}{\left({x}\right)}},\ldots{f{{\left({x}\right)}}}={e}^{{{2}{x}}}\)

Answers (1)

2021-06-28

\(\displaystyle{f{{\left({x}\right)}}}={e}^{{{2}{x}}}\) Find f'(x), differentiate both sides with respect to x
\(\displaystyle{f}'{\left({x}\right)}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{e}^{{{2}{x}}}\right]}\)
\(\displaystyle{f}'{\left({x}\right)}={2}{e}^{{{2}{x}}}\)
Find f''(x)
\(\displaystyle{f}{''}{\left({x}\right)}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{f}'{\left({x}\right)}\right]}\)
\(\displaystyle{f}{''}{x}{)}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{2}{e}^{{{2}{x}}}\right]}\)
\(f''(x)=4e^{2x}\)
\(\displaystyle{f}{''}{\left({x}\right)}={2}^{{{2}}}{e}^{{{2}{x}}}\)
Find f'''(x) \(\displaystyle{f}{'''}{\left({x}\right)}={8}{e}^{{{2}{x}}}\)
\(\displaystyle{f}{'''}{\left({x}\right)}={2}^{{{3}}}{e}^{{{2}{x}}}\)
Find \(f^{4}(x)\)
\(\displaystyle{{f}^{{{\left({4}\right)}}}{\left({x}\right)}}={2}^{{{4}}}{e}^{{{2}{x}}}\)
\(\displaystyle{{f}^{{{\left({4}\right)}}}{\left({x}\right)}}={16}{e}^{{{2}{x}}}\)
Therefore,
\(\displaystyle{{f}^{{{n}}}{\left({x}\right)}}={2}^{{{n}}}{e}^{{{2}{x}}}\)

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