Problem 22 Differentiate. $$G(x)=-7 e^{-x... [FREE SOLUTION] (2024)

Short Answer

Exponential Functions

Exponential functions are a type of mathematical function that involve the constant base of the natural logarithm, e, raised to a power that is a variable. The general form of an exponential function is given by: \[f(x) = e^{g(x)}\].
In these types of functions, the base e (approximately 2.71828) is a constant, and the exponent is a function of the variable x. Exponential functions are crucial in many areas of mathematics and science because they describe growth and decay processes, among other phenomena.
When differentiating or integrating these functions, specific rules apply because the rate of change of an exponential function is proportional to the value of the function itself. This property makes them relatively straightforward to differentiate.
For example, if we have the exponential function \(e^{-x}\), the process involves applying the chain rule, which helps us differentiate complex functions. This concept is important to understand and use effectively when handling exponential functions.

Constant Multiple Rule

The constant multiple rule is an essential principle in differentiation. It states that if a function is multiplied by a constant, the derivative of this function is the constant multiplied by the derivative of the original function. Mathematically, it can be expressed as:
\[ \frac{d}{dx} [c \times f(x)] = c \times \frac{d}{dx} [f(x)] \],
where \(c\) is a constant.
In our example with the function \(G(x) = -7 e^{-x}\), we see that -7 is the constant. When differentiating \(e^{-x}\), the constant -7 remains in front and is multiplied by the derivative of \(e^{-x}\).
Understanding this rule simplifies the process of differentiation significantly, as it allows us to deal with the constants separately and focus on differentiating the variable terms properly. This rule is especially useful when handling functions like polynomials and exponential functions that are multiplied by constant coefficients.

Chain Rule

The Chain Rule is a fundamental theorem for differentiating composite functions. When a function can be seen as a composition of two or more functions, the chain rule helps to differentiate it effectively. The rule states:
\[ \frac{d}{dx} [f(g(x))] = f'(g(x)) \times g'(x) \],
which means that to differentiate a composite function, you need to multiply the derivative of the outer function evaluated at the inner function by the derivative of the inner function.
In the context of our example, when differentiating \(e^{-x}\), we apply the chain rule because \(-x\) is the inner function and \(e^{u}\) with \(u = -x\) is the outer function. The differentiation process involves:
1. Taking the derivative of the outer function:\( e^{u} \text{ with respect to } u\) which remains \(e^{u}\).2. Next, differentiate the inner function \(-x\) with respect to \(x\), giving a derivative of \(-1\).3. Finally, multiply these derivatives together to get the derivative of the composite function as \(-e^{-x}\).
This rule is one of the cornerstones of calculus, enabling us to break down and differentiate more complicated expressions step by step.