Find the Derivative – d/dx 10xe^(-4x)

$10 x e^{- 4 x}$

Since $10$ is constant with respect to $x$ , the derivative of $10 x e^{- 4 x}$ with respect to $x$ is $10 \frac{d}{d x} [x e^{- 4 x}]$ .

$10 \frac{d}{d x} [x e^{- 4 x}]$

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{- 4 x}$ .

$10 (x \frac{d}{d x} [e^{- 4 x}] + e^{- 4 x} \frac{d}{d x} [x])$

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - 4 x$ .

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To apply the Chain Rule, set $u$ as $- 4 x$ .

$10 (x (\frac{d}{d u} [e^{u}] \frac{d}{d x} [- 4 x]) + e^{- 4 x} \frac{d}{d x} [x])$

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$10 (x (e^{u} \frac{d}{d x} [- 4 x]) + e^{- 4 x} \frac{d}{d x} [x])$

Replace all occurrences of $u$ with $- 4 x$ .

$10 (x (e^{- 4 x} \frac{d}{d x} [- 4 x]) + e^{- 4 x} \frac{d}{d x} [x])$

Differentiate.

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Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$10 (x e^{- 4 x} (- 4 \frac{d}{d x} [x]) + e^{- 4 x} \frac{d}{d x} [x])$

Move $- 4$ to the left of $x e^{- 4 x}$ .

$10 (- 4 \cdot (x e^{- 4 x}) \frac{d}{d x} [x] + e^{- 4 x} \frac{d}{d x} [x])$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$10 (- 4 x e^{- 4 x} \cdot 1 + e^{- 4 x} \frac{d}{d x} [x])$

Multiply $- 4$ by $1$ .

$10 (- 4 x e^{- 4 x} + e^{- 4 x} \frac{d}{d x} [x])$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$10 (- 4 x e^{- 4 x} + e^{- 4 x} \cdot 1)$

Multiply $e^{- 4 x}$ by $1$ .

$10 (- 4 x e^{- 4 x} + e^{- 4 x})$

Simplify.

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Apply the distributive property.

$10 (- 4 x e^{- 4 x}) + 10 e^{- 4 x}$

Multiply $- 4$ by $10$ .

$- 40 x e^{- 4 x} + 10 e^{- 4 x}$

Reorder terms.

$- 40 e^{- 4 x} x + 10 e^{- 4 x}$

Find the Derivative – d/dx 10xe^(-4x)

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