Question

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

Derivatives
ANSWERED
asked 2021-06-04
Find the derivatives of the functions. \(\displaystyle{t}{\left({x}\right)}={3}^{{{2}{x}-{3}}}\)

Answers (1)

2021-06-05
Recall the generalized derivative rule for exponential functions:
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{b}^{{{u}}}\right]}={b}^{{{u}}}{\ln{{b}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}\)
In this exercise, we want to find the derivative of
\(\displaystyle{t}{\left({x}\right)}={3}^{{{2}{x}-{4}}}\)
To make use of the rule, denote the argument of the exponential function by u, i.e.
u=2x-4
Using the rule from Step 1, we get:
\(\displaystyle{t}'{\left({x}\right)}={3}^{{{u}}}{\ln{{3}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}={3}^{{{2}{x}-{4}}}{\ln{{3}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({2}{x}-{4}\right)}\rbrace={2}\)
\(\displaystyle{2}\dot{{\lbrace}}{3}^{{{2}{x}-{4}}}{\ln{{3}}}\)
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