Question

Find the derivatives of the functions. g(x)=e^{3x-1}e^{x-2}e^{x}

Derivatives
Find the derivatives of the functions. $$\displaystyle{g{{\left({x}\right)}}}={e}^{{{3}{x}-{1}}}{e}^{{{x}-{2}}}{e}^{{{x}}}$$

2021-06-27
The function we want to differentiate is
$$\displaystyle{g{{\left({x}\right)}}}={e}^{{{3}{x}-{1}}}{e}^{{{x}-{2}}}{e}^{{{x}}}$$
Apply the exponential property
$$\displaystyle{e}^{{{a}}}{e}^{{{b}}}={e}^{{{a}+{b}}}$$
to get
$$\displaystyle{g{{\left({x}\right)}}}={e}^{{{3}{x}-{1}+{x}-{2}+{x}}}={e}^{{{5}{x}-{3}}}$$
Now we can use the generalized rule for the derivative of exponential function:
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({e}^{{{u}}}\right)}={e}^{{{u}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}$$
We get
$$\displaystyle{g}'{\left({x}\right)}={e}^{{{5}{x}-{3}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({5}{x}-{3}\right)}={5}{e}^{{{5}{x}-{3}}}$$