"Find the derivatives of the functions. t(x)=4−x+5" was answered by our community

College
Calculus and Analysis
Differential Calculus
Derivatives

Khadija Wells

Answered question

2021-07-05

Find the derivatives of the functions.

t (x) = 4^{- x + 5}

Answer & Explanation

Yusuf Keller

Skilled2021-07-06Added 90 answers

Recall the generalized derivative rule for exponential functions:
$\frac{d}{d x} [b^{u}] = b^{u} \ln b \frac{d u}{d x}$
In this exercise, we want to find the derivative of
$t (x) = 4^{- x + 5}$
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:
$t^{'} (x) = 4^{u} \ln 4 \frac{d u}{d x} = 4^{- x + 5} \ln 3 \frac{d}{d x} (- x + 5)} = - 1$
$= - 4^{- x + 5} \ln 4$

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