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Magazine Chapter 3: Problem 33
Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ g(\alpha)=e^{\alpha e^{-2 \alpha}} $$
Short Answer
Expert verified
The derivative is \( \frac{dg}{d\alpha} = e^{\alpha e^{-2\alpha}} e^{-2\alpha} (1 - 2\alpha) \).
Step by step solution
01
Identify the function structure
The given function is \( g(\alpha) = e^{\alpha e^{-2\alpha}} \). This is a composite function, and we need to use the chain rule to find the derivative.
02
Apply the chain rule
Let \( u = \alpha e^{-2\alpha} \). Then \( g(\alpha) = e^u \). The derivative of \( e^u \) with respect to \( \alpha \) is \( e^u \cdot \frac{du}{d\alpha} \). We need to find \( \frac{du}{d\alpha} \).
03
Differentiate \(u\) using the product rule
The expression for \( u \) is \( \alpha e^{-2\alpha} \). This is a product of two functions: \( \alpha \) and \( e^{-2\alpha} \). Use the product rule: \( \frac{d}{d\alpha}(fg) = f'g + fg' \). Here, \( f = \alpha \) and \( g = e^{-2\alpha} \).
04
Calculate derivatives of \(f\) and \(g\)
Calculate: - \( \frac{d}{d\alpha}(\alpha) = 1 \) - \( \frac{d}{d\alpha}(e^{-2\alpha}) = e^{-2\alpha} \cdot (-2) \) using the chain rule for the exponent.
05
Apply the product rule
Using the product rule, \( \frac{du}{d\alpha} = 1 \cdot e^{-2\alpha} + \alpha \cdot \left(e^{-2\alpha} \cdot (-2)\right) \). Simplify this to get \( e^{-2\alpha} - 2\alpha e^{-2\alpha} \).
06
Find the derivative of \(g(\alpha)\)
Substitute \( u \) and \( \frac{du}{d\alpha} \) back into the derivative of \( g(\alpha) \), which is \( e^u \cdot \frac{du}{d\alpha} \). Therefore, \( \frac{dg}{d\alpha} = e^{\alpha e^{-2\alpha}}(e^{-2\alpha} - 2\alpha e^{-2\alpha}) \).
07
Simplify the expression
The simplified expression of the derivative is \( \frac{dg}{d\alpha} = e^{\alpha e^{-2\alpha}} e^{-2\alpha} (1 - 2\alpha) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chain Rule
When finding derivatives, especially for composite functions, the chain rule is a powerful tool that simplifies this process. The chain rule is applicable when you have a function composed of another function, such as in the case of an exponential function with a variable inside it. To apply the chain rule, follow these steps:
- Identify the outer function and the inner function. In our example, the outer function is the exponential function, and the inner function is the exponent itself.
- Take the derivative of the outer function with respect to the inner function.
- Multiply by the derivative of the inner function with respect to the original variable.
Product Rule
The product rule is essential when dealing with the derivative of a product of two functions. When a function can be expressed as a product, say \( u(x) \) and \( v(x) \), its derivative is given by:
\[ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \]
This means you:
\[ \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \]
This means you:
- Differentiate the first function, leaving the second function unchanged.
- Differentiate the second function, leaving the first function unchanged.
- Add the two products from the previous steps.
Composite Function
A composite function is essentially a function of a function. It involves applying one function to the results of another. For example, in the expression \( e^{\alpha e^{-2\alpha}} \), the exponent \( \alpha e^{-2\alpha} \) itself is a function. Recognizing and simplifying such composite structures is key to solving derivative problems efficiently. Here鈥檚 a detailed breakdown on how to handle them:
- Identify each level of the function structure, particularly focusing on which part depends on what inside the function.
- Break down the function into its components, beginning with the inner-most function and working outward, often making use of substituting variables.
- Apply appropriate derivative rules, like the chain rule or product rule, to each component respectively.
Exponential Function
Exponential functions are a staple in calculus due to their unique property: their rate of change is proportional to their current value, which simplifies derivative calculations. An exponential function generally takes the form \( e^u \), where \( u \) is often another function of the variable.
To differentiate \( e^u \) with respect to the variable, the derivative is \( e^u \times \frac{du}{dx} \) using the chain rule.
To differentiate \( e^u \) with respect to the variable, the derivative is \( e^u \times \frac{du}{dx} \) using the chain rule.
- This simplifies problems, as the exponential function remains in the differentiation process.
- This property allows for elegant solutions in many real-world problems involving growth and decay.
