Step by step solution to "Derivatives of products and quotients Find the derivative..." - Plainmath

Sherry Becker

Answered question

2021-11-28

Derivatives of products and quotients Find the derivative of the following functions by first expanding or simplifying the expression.

r (t) = \frac{e^{2 t} + 3 e^{t} + 2}{e^{t} + 2}

Answer & Explanation

Muspee

Beginner2021-11-29Added 13 answers

Given,

r (t) = \frac{e^{2 t} + 3 e^{t} + 2}{e^{t} + 2}

\Rightarrow r (t) = \frac{e^{2 t} + 2 e^{t} + e^{t} + 2}{e^{t} + 2}

\Rightarrow r (t) = \frac{e^{t} (e^{t} + 2) + (e^{t} + 2)}{e^{t} + 2}

\Rightarrow r (t) = \frac{(e^{t} + 2) (e^{t} + 1)}{e^{t} + 2}

\Rightarrow r (t) = e^{t} + 1

Now differentiating with respect to t, we get

r^{'} (t) = e^{t} + 0 [∵ \frac{d (e^{x})}{d x} = e^{x}, \frac{d (c)}{d x} = 0, c \in R]

\Rightarrow r^{'} (t) = e^{t}

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