91Ó°ÊÓ

The chain rule is a fundamental technique in calculus for differentiating composite functions. To apply the chain rule, identify the outer function and inner function in a composite function.
For example, in the function \( -e^{-3x} \), the outer function is the exponential function \( e^u \), where \( u = -3x \) is the inner function. The chain rule states that the derivative of a composite function \( f(g(x)) \) is given by:
\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \times g'(x) \]
This means you first find the derivative of the outer function with respect to the inner function, then multiply it by the derivative of the inner function with respect to \( x \). In our example, the outer derivative is \( -e^{-3x} \) and the inner derivative is \( -3 \). Thus, using the chain rule, the derivative is \( 3e^{-3x} \).

Exponential functions have the form \( e^{u} \), where \( e \) is the base of natural logarithms, approximately equal to 2.71828. These functions are special because their derivatives are proportional to the original function.
In simpler terms, differentiating \( e^{u} \) results in \( e^{u} \), but if \( u \) itself is a function of \( x \), we must also apply the chain rule.
For example, when differentiating the function \( -e^{-3x} \), \( u = -3x \) inside the exponent influences the outcome. As a result, you use the chain rule to get \( \frac{d}{dx} -e^{-3x} = -e^{-3x} \times (-3) = 3e^{-3x} \). Exponential functions grow or decay quickly, making them vital for modeling growth processes.

Differentiating a function involves several steps to ensure you accurately find the derivative. In the given problem, \( y = 1 - e^{-3x} \), the steps are:

1. **Identify the components:** Recognize that the function has two parts: a constant \( 1 \) and an exponential function \( -e^{-3x} \).
2. **Differentiate the constant:** The derivative of a constant is always zero because constants do not change.
3. **Differentiate the exponential function using the chain rule:** Let \( u = -3x \). The derivative of \( e^u \) with respect to \( u \) is \( e^u \), then multiply by \( \frac{du}{dx} = -3 \). This gives \( -e^{-3x} \times -3 = 3e^{-3x} \).
4. **Combine the results:** The constant part contributes \( 0 \), and the exponential part contributes \( 3e^{-3x} \). Thus, the final derivative of the function is \( \frac{dy}{dx} = 3e^{-3x} \).
Breaking down the process helps you understand each aspect and apply differentiation techniques effectively.