"Find the derivatives of the functions y=(1+2x)e−2x" was answered by our experts

abondantQ

Answered question

2021-10-07

Find the derivatives of the functions

y = (1 + 2 x) e^{- 2 x}

Answer & Explanation

Raheem Donnelly

Skilled2021-10-08Added 75 answers

Given,
$y = (1 + 2 x) e^{- 2 x}$
Differentiating with respect to x, we get
$y = (1 + 2 x) e^{- 2 x}$
$\frac{d y}{d x} = \frac{d (1 + 2 x)}{d x} \cdot e^{- 2 x} + \frac{d (e^{- 2 x})}{d x} \cdot (1 + 2 x) [∵ \frac{d (u v)}{d x} = \frac{d u}{d x} \cdot v + \frac{d v}{d x} \cdot u]$
$\Rightarrow \frac{d y}{d x} = [0 + 2 (1)] \cdot e^{- 2 x} + e^{- 2 x} \cdot \frac{d (- 2 x)}{d x} \cdot (1 + 2 x) [\begin{matrix} ∵ \frac{d (c)}{d x} = 0, c \in R ∖ \frac{d (x^{n})}{d x} = n x^{n - 1} \frac{d (e^{x})}{d x} = e^{x} \end{matrix}]$
$\Rightarrow \frac{d y}{d x} = 2 e^{- 2 x} + e^{- 2 x} \cdot (- 2) \cdot (1 + 2 x)$
$\Rightarrow \frac{d y}{d x} = 2 e^{- 2 x} - 2 e^{- 2 x} \cdot (1 + 2 x)$
$\Rightarrow \frac{d y}{d x} = 2 e^{- 2 x} (1 - (1 + 2 x))$
$\Rightarrow \frac{d y}{d x} = - 4 x e^{- 2 x}$

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