How do you differentiate e^(-10x) ?

To differentiate \( e^{-10x} \), we will use the chain rule of differentiation. Here's a step-by-step explanation:

1. **Understand the chain rule**:
The chain rule states that if a function \( y \) is composed of an outer function \( u \) and an inner function \( v \), such that \( y = u(v(x)) \), then the derivative of \( y \) with respect to \( x \) is:
\[
\frac{dy}{dx} = \frac{du}{dv} \cdot \frac{dv}{dx}.
\]

2. **Identify the components**:
In \( e^{-10x} \):
- The outer function \( u(v) = e^v \) (the exponential function).
- The inner function \( v(x) = -10x \) (a linear function).

3. **Differentiate the outer function**:
The derivative of \( e^v \) with respect to \( v \) is \( e^v \).

4. **Differentiate the inner function**:
The derivative of \( v(x) = -10x \) with respect to \( x \) is \( -10 \).

5. **Apply the chain rule**:
Using the chain rule:
\[
\frac{d}{dx}\left(e^{-10x}\right) = \frac{d}{dv}(e^v) \cdot \frac{dv}{dx}.
\]
Substituting \( v = -10x \):
\[
\frac{d}{dx}\left(e^{-10x}\right) = e^{-10x} \cdot (-10).
\]

6. **Simplify the expression**:
\[
\frac{d}{dx}\left(e^{-10x}\right) = -10e^{-10x}.
\]

### Final Answer:
The derivative of \( e^{-10x} \) with respect to \( x \) is:
\[
\frac{d}{dx}\left(e^{-10x}\right) = -10e^{-10x}.
\]