Find the Derivative – d/dx y=e^(-7x)

$y = e^{- 7 x}$

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 $g (x) = - 7 x$ .

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To apply the Chain Rule, set $u$ as $- 7 x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [- 7 x]$

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

$e^{u} \frac{d}{d x} [- 7 x]$

Replace all occurrences of $u$ with $- 7 x$ .

$e^{- 7 x} \frac{d}{d x} [- 7 x]$

Differentiate.

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Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$e^{- 7 x} (- 7 \frac{d}{d x} [x])$

Move $- 7$ to the left of $e^{- 7 x}$ .

$- 7 \cdot e^{- 7 x} \frac{d}{d x} [x]$

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

$- 7 e^{- 7 x} \cdot 1$

Multiply $- 7$ by $1$ .

$- 7 e^{- 7 x}$

Find the Derivative – d/dx y=e^(-7x)

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