Question

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

Derivatives
Calculate the derivatives of all orders:
$$\displaystyle{f}'{\left({x}\right)},{f}{''}{\left({x}\right)},{f}{'''}{\left({x}\right)},{{f}^{{{4}}}{\left({x}\right)}},\ldots{{f}^{{{n}}}{\left({x}\right)}},\ldots{f{{\left({x}\right)}}}={e}^{{{2}{x}}}$$

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,
$$\displaystyle{{f}^{{{n}}}{\left({x}\right)}}={2}^{{{n}}}{e}^{{{2}{x}}}$$