Question

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

Derivatives
ANSWERED
asked 2021-07-05
Find the derivatives of the functions. \(\displaystyle{t}{\left({x}\right)}={4}^{{-{x}+{5}}}\)

Answers (1)

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:
\(\displaystyle{t}'{\left({x}\right)}={4}^{{{u}}}{\ln{{4}}}{\frac{{{d}{u}}}{{{\left.{d}{x}\right.}}}}={4}^{{-{x}+{5}}}{\ln{{3}}}{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left(-{x}+{5}\right)}\rbrace=-{1}\)
\(\displaystyle=-{4}^{{-{x}+{5}}}{\ln{{4}}}\)

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