Question

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

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

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:
$$\displaystyle{t}'{\left({x}\right)}={3}^{{{u}}}{\ln{{3}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}={3}^{{{2}{x}-{4}}}{\ln{{3}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({2}{x}-{4}\right)}\rbrace={2}$$
$$\displaystyle{2}\dot{{\lbrace}}{3}^{{{2}{x}-{4}}}{\ln{{3}}}$$