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Magazine Chapter 4: Problem 13
Differentiate. $$ f(x)=5 e^{-2 x} $$
Short Answer
Expert verified
The derivative of \(f(x) = 5 e^{-2x}\) is \(-10 e^{-2x}\).
Step by step solution
01
- Identify the function components
The given function is: \[ f(x) = 5 e^{-2x} \] This function is made up of the constant 5 and the exponential function \(e^{-2x}\), where -2x is the exponent.
02
- Apply the chain rule
To differentiate \( f(x) = 5 e^{-2x} \), use the chain rule. Recall that the chain rule states: \[ \frac{d}{dx}[g(h(x))] = g'(h(x)) \times h'(x) \] Here, g(x) = e^x, and h(x) = -2x.
03
- Differentiate the outer function
First, differentiate the outer function with respect to the inner function. The derivative of \( e^{u} \) with respect to \( u \) is \( e^{u} \). So, the derivative of \( e^{-2x} \) with respect to \( -2x \) is: \[ \frac{d}{du}[e^{u}] = e^{u} \] Applying this, we get: \[ \frac{d}{d(-2x)}[e^{-2x}] = e^{-2x} \]
04
- Differentiate the inner function
Now differentiate the inner function \( -2x \) with respect to \( x \): \[ \frac{d}{dx}[-2x] = -2 \]
05
- Combine the results
Multiply the results from Steps 3 and 4: \[ \frac{d}{dx}[5 e^{-2x}] = 5 \cdot e^{-2x} \cdot (-2) \] Simplifying this gives: \[ \frac{d}{dx}[5 e^{-2x}] = -10 e^{-2x} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Chain Rule
The chain rule is a fundamental concept in calculus. It's used for finding the derivative of a composite function. Let's break this down simply.
The chain rule states that if you have a function composed of other functions, like \( f(g(x)) \), you differentiate the outer function first and then multiply by the derivative of the inner function.
In our exercise, we have \( f(x) = 5e^{-2x} \). Here, \( g(x) = e^x \) is the outer function and \( h(x) = -2x \) is the inner function.
Applying the chain rule helps you smoothly differentiate such composite functions by taking it step by step.
The chain rule states that if you have a function composed of other functions, like \( f(g(x)) \), you differentiate the outer function first and then multiply by the derivative of the inner function.
In our exercise, we have \( f(x) = 5e^{-2x} \). Here, \( g(x) = e^x \) is the outer function and \( h(x) = -2x \) is the inner function.
Applying the chain rule helps you smoothly differentiate such composite functions by taking it step by step.
Exponential Functions and Their Derivatives
An exponential function has the form \( e^x \), where \( e \) is a mathematical constant approximately equal to 2.71828. These functions grow rapidly and are crucial in many fields, from biology to finance.
For our problem, we need to differentiate an exponential function with a more complex exponent, \( e^{-2x} \), which means applying the chain rule.
The derivative of \( e^u \) with respect to \( u \) is still \( e^u \). Thus, for \( e^{-2x} \), the derivative with respect to \( -2x \) remains \( e^{-2x} \), capturing the essence of exponential growth even when transformed by a negative exponent.
For our problem, we need to differentiate an exponential function with a more complex exponent, \( e^{-2x} \), which means applying the chain rule.
The derivative of \( e^u \) with respect to \( u \) is still \( e^u \). Thus, for \( e^{-2x} \), the derivative with respect to \( -2x \) remains \( e^{-2x} \), capturing the essence of exponential growth even when transformed by a negative exponent.
Basics of Calculus
Calculus is all about change. When we differentiate a function, we are looking for how it changes in relation to its variable. This is known as finding the derivative.
In simpler terms, the derivative tells us the rate of change or the slope of the function at any given point.
In our example, differentiating \( f(x) = 5e^{-2x} \) involves both the constant multiple rule and chain rule. The constant multiplier (in this case, 5) remains as is, while we apply the chain rule to the exponential part.
In simpler terms, the derivative tells us the rate of change or the slope of the function at any given point.
In our example, differentiating \( f(x) = 5e^{-2x} \) involves both the constant multiple rule and chain rule. The constant multiplier (in this case, 5) remains as is, while we apply the chain rule to the exponential part.
Understanding Derivatives
Derivatives are a core part of calculus. They represent the rate at which a function is changing. The process of finding a derivative is called differentiation.
For the function \( f(x) = 5e^{-2x} \), differentiation involves using the chain rule and product rule. First, we treat the 5 as a constant and focus on differentiating \( e^{-2x} \).
As derived, the derivative of the outer function, \( e^{-2x} \), with respect to its inner function \(-2x\) remains \( e^{-2x} \).
Then, differentiating the inner function \(-2x\) with respect to \( x \) gives us \(-2\).
Combining these results using the product rule, we get the final derivative as: \( f'(x) = -10e^{-2x} \).
For the function \( f(x) = 5e^{-2x} \), differentiation involves using the chain rule and product rule. First, we treat the 5 as a constant and focus on differentiating \( e^{-2x} \).
As derived, the derivative of the outer function, \( e^{-2x} \), with respect to its inner function \(-2x\) remains \( e^{-2x} \).
Then, differentiating the inner function \(-2x\) with respect to \( x \) gives us \(-2\).
Combining these results using the product rule, we get the final derivative as: \( f'(x) = -10e^{-2x} \).
