Question

# Find the derivatives of the functions. t(x)=4^{-x+5}

Derivatives
Find the derivatives of the functions. $$\displaystyle{t}{\left({x}\right)}={4}^{{-{x}+{5}}}$$

2021-07-06

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
$$\displaystyle{t}{\left({x}\right)}={4}^{{-{x}+{5}}}$$
To make use of the rule, denote the argument of the exponential function by u,i.e.
$$u=-x+5$$
Using the rule from Step 1, we get:
$$\displaystyle{t}'{\left({x}\right)}={4}^{{{u}}}{\ln{{4}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}={4}^{{-{x}+{5}}}{\ln{{3}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left(-{x}+{5}\right)}\rbrace=-{1}$$
$$\displaystyle=-{4}^{{-{x}+{5}}}{\ln{{4}}}$$