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Magazine Chapter 3: Problem 39
Differentiate. $$ f(x)=e^{-x^{2}+8 x} $$
Short Answer
Expert verified
The derivative is \( f'(x) = (8 - 2x) e^{-x^2 + 8x} \).
Step by step solution
01
Identify the Function
The function we are dealing with is an exponential function: \[ f(x) = e^{-x^2 + 8x} \].Our goal is to differentiate this function with respect to \( x \).
02
Recognize the Chain Rule
Since \( f(x) = e^{u(x)} \) where \( u(x) = -x^2 + 8x \), we must use the chain rule to differentiate.The chain rule states that \[ \frac{d}{dx}[e^{u(x)}] = e^{u(x)} \frac{du}{dx} \].
03
Differentiate the Inner Function
First, differentiate the inner function \( u(x) = -x^2 + 8x \) with respect to \( x \):- The derivative of \(-x^2\) is \(-2x\).- The derivative of \(8x\) is \(8\).Thus, the derivative \( \frac{du}{dx} = -2x + 8 \).
04
Apply the Chain Rule
Now apply the chain rule:\[ \frac{d}{dx}[f(x)] = e^{u(x)} \cdot \frac{du}{dx} \]Substitute \( u(x) = -x^2 + 8x \) and \( \frac{du}{dx} = -2x + 8 \) into the equation:\[ f'(x) = e^{-x^2 + 8x} \cdot (-2x + 8) \].
05
Simplify the Expression
The derivative of the function is:\[ f'(x) = (8 - 2x) \cdot e^{-x^2 + 8x} \].This is the final, simplified form of the derivative.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chain Rule
When differentiating complex functions, the chain rule is your best friend. It helps break down the process into simpler, more manageable parts. Consider you have a composition of functions, where a function is applied inside another. In our exercise, we had the exponential function \( f(x) = e^{-x^2 + 8x} \). Here, the power \( -x^2 + 8x \) is its own function, which we refer to as \( u(x) \).
This method is powerful because it allows you to deal with complicated expressions separately, making differentiation easier and less error-prone.
- The chain rule states that to find the derivative of a composite function \( f(u(x)) \), we need to multiply the derivative of the outer function, \( f \), evaluated at \( u(x) \), by the derivative of the inner function, \( u(x) \).
- In mathematical terms, if \( f(x) = e^{u(x)} \), then \[ \frac{d}{dx}[e^{u(x)}] = e^{u(x)} \cdot \frac{du}{dx} \]
This method is powerful because it allows you to deal with complicated expressions separately, making differentiation easier and less error-prone.
Derivative Calculation
Calculating derivatives involves understanding how a function changes. More formally, it tells us the rate at which a function's value is changing at any given point. For the function power \( u(x) = -x^2 + 8x \) in our exercise, we had to find its derivative first.
Combining these results, the derivative of \( u(x) \) is \( \frac{du}{dx} = -2x + 8 \).
- Breaking it down, differentiate \( -x^2 \) which results in \( -2x \).
- Then, differentiate \( 8x \) which gives \( 8 \).
Combining these results, the derivative of \( u(x) \) is \( \frac{du}{dx} = -2x + 8 \).
- Now we apply this in the chain rule to find \( f'(x) \). We multiply \( e^{-x^2 + 8x} \) by the derivative \( -2x + 8 \).
- Thus, \[ f'(x) = e^{-x^2 + 8x} \times (-2x + 8) \]
- Simplifying further, we get the final result: \( f'(x) = (8 - 2x) \cdot e^{-x^2 + 8x} \).
Exponential Function
An exponential function is characterized by its constant base, here \( e \), which is the natural logarithm base, and its variable exponent. This makes the function exhibit a rapid growth or decay based on the sign of the exponent. In this task, our main function was \( e^{-x^2 + 8x} \).
By understanding the behavior of \( e \) and applying differentiation techniques accurately, you can successfully tackle problems involving exponential growth or decay in various fields like economics, physics, and statistics.
- Note that \( e \) raised to any power is never zero, implying that exponential functions produce positive outputs, which makes them quite interesting to work with.
- The derivative of \( e^x \) is \( e^x \) itself, but when coupled with the chain rule in more complex exponentials like \( e^{-x^2 + 8x} \), the differentiation becomes a bit more involved.
By understanding the behavior of \( e \) and applying differentiation techniques accurately, you can successfully tackle problems involving exponential growth or decay in various fields like economics, physics, and statistics.
