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Chapter 4: Problem 18

Differentiate. $$ f(x)=-4 e^{x} $$

Short Answer

Expert verified
The derivative is \( f'(x) = -4 e^x \).

Step by step solution

01

Identify the function to be differentiated

Given the function to differentiate: \[ f(x) = -4 e^x \] Notice that this is a simple exponential function with a constant multiple.
02

Apply the differentiation rule for exponential functions

The differentiation rule for the exponential function \( e^x \) is: \[ \frac{d}{dx} (e^x) = e^x \] Since our function is multiplied by a constant -4, this constant remains unchanged during differentiation.
03

Differentiate the function

Differentiate \( f(x) = -4 e^x \) using the product rule for constants: \[ \frac{d}{dx} (-4 e^x) = -4 \cdot \frac{d}{dx} (e^x) = -4 e^x \] So, the derivative is: \[ f'(x) = -4 e^x \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponential Function
An exponential function is a type of mathematical function that grows rapidly. Its general form is \text{f(x) = a} e^{bx}, where \text{a} and \text{b} are constants, and \( e^x \) is the natural exponential function, also written as exp(x). In our example, the specific exponential function is f(x) = -4 e^x. Exponential functions are unique because their rate of growth is proportional to their current value.
For instance, the natural exponential function, \( e^x \), has a base of 'e,' an irrational number approximately equal to 2.71828.
Exponential functions are widely used in various fields like population growth, finance, and natural sciences due to their properties.
Differentiation Rules
In calculus, differentiation is the process of finding the rate at which a function changes. The differentiation rule for exponential functions is particularly straightforward. For the natural exponential function, \( e^x \), the derivative is itself: \text{d/dx} (e^x) = e^x.
Here are some differentiation rules to remember:
  • The Power Rule: \text{d/dx} (x^n) = nx^(n-1).
  • The Constant Rule: \text{d/dx} (c) = 0, where c is a constant.
  • The Sum Rule: \text{d/dx} (f(x) + g(x)) = f'(x) + g'(x).
  • The Product Rule: \text{d/dx} (u \text{v}) = u' v + u \text{v}' (for functions u and v).
This particular problem uses the rule for differentiating exponential functions, where the function 'e^x' remains the same after differentiation.
Constant Multiple
When differentiating functions, if a constant is multiplied by a function, the constant remains unchanged. This is known as the constant multiple rule.
The rule can be stated as: If \(C\) is a constant and \(f(x)\) is a differentiable function, then: \text{d/dx} [C f(x)] = C \text{f'(x)}.
In our example, our function is f(x) = -4 e^x. Here, -4 is a constant multiple.
Applying the constant multiple rule, we get: \text{d/dx} (-4 e^x) = -4 \text{d/dx} (e^x) = -4 e^x.
As you can see, the constant -4 persists after differentiation, and the differentiation of 'e^x' remains 'e^x'. This rule is simple but essential for solving differentiation problems involving constants.

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