Question

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

Derivatives
ANSWERED
asked 2021-06-26
Find the derivatives of the functions. \(\displaystyle{g{{\left({x}\right)}}}={e}^{{{3}{x}-{1}}}{e}^{{{x}-{2}}}{e}^{{{x}}}\)

Answers (1)

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
\(\displaystyle{g}'{\left({x}\right)}={e}^{{{5}{x}-{3}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({5}{x}-{3}\right)}={5}{e}^{{{5}{x}-{3}}}\)
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