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Magazine Chapter 4: Problem 11
Find the derivative. \(x^{5} e^{x}\)
Short Answer
Expert verified
The derivative is \(x^4 e^x (5+x)\).
Step by step solution
01
Identify the Functions
The given function is a product of two functions, namely, \(f(x) = x^5\) and \(g(x) = e^x\).
02
Use the Product Rule
The derivative of a product of two functions, \(f(x)\) and \(g(x)\), is given by the product rule: \((f\cdot g)' = f'\cdot g + f\cdot g'\).
03
Find the Derivative of \(f(x) = x^5\)
The derivative of \(f(x) = x^5\) is \(f'(x) = 5x^4\) using the power rule, which states that \(\frac{d}{dx}(x^n) = nx^{n-1}\).
04
Find the Derivative of \(g(x) = e^x\)
The derivative of \(g(x) = e^x\) is \(g'(x) = e^x\), as the derivative of an exponential function \(e^x\) is itself, \(e^x\).
05
Apply the Product Rule
Substitute the derivatives found in Steps 3 and 4 into the product rule: \[ (x^5 e^x)' = (5x^4)(e^x) + (x^5)(e^x) \]
06
Simplify the Expression
The derivative simplifies to: \[ 5x^4 e^x + x^5 e^x = x^4 e^x (5 + x) \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Product Rule
When you're dealing with a function that is the product of two other functions, like in our exercise, the product rule is handy. The product rule helps find the derivative of the product of two functions. Let's break this down:
In our exercise's case, we used functions \( f(x) = x^5 \) and \( g(x) = e^x \). This approach made it straightforward to apply the formula and calculate the derivative.
- If you have two functions, say \( f(x) \) and \( g(x) \), their product is \( f(x) \cdot g(x) \).
- The product rule formula states: \((f\cdot g)' = f'\cdot g + f\cdot g'\).
- This means to find the derivative of their product, you take the derivative of the first one multiplied by the second as is, plus the first function as is, times the derivative of the second.
In our exercise's case, we used functions \( f(x) = x^5 \) and \( g(x) = e^x \). This approach made it straightforward to apply the formula and calculate the derivative.
Power Rule
The power rule is a simple and essential tool in calculus. It's a straightforward method to find the derivative of a function that is a power of \( x \). Here's how it works:
This step is crucial and needs to be addressed accurately to ensure the correctness of the entire derivative process. Understanding and applying the power rule will make your calculus journey easier and more intuitive.
- For a function of the form \( x^n \), its derivative is given by \( nx^{n-1} \).
- This means you multiply the power by the function itself and then decrease the power by one.
This step is crucial and needs to be addressed accurately to ensure the correctness of the entire derivative process. Understanding and applying the power rule will make your calculus journey easier and more intuitive.
Exponential Function
Exponential functions like \( e^x \) have unique properties, especially when differentiating. Let's explore what makes differentiating exponential functions straightforward and interesting:
Understanding this property of exponential functions is powerful, helping simplify much of your work in calculus and providing a clear and consistent path for taking derivatives.
- The function \( e^x \) is especially significant because its derivative is itself, that is, \( \frac{d}{dx}(e^x) = e^x \).
- This property makes working with \( e^x \) very simple, as you don't have to worry about complex adjustments or changes when finding its derivative.
Understanding this property of exponential functions is powerful, helping simplify much of your work in calculus and providing a clear and consistent path for taking derivatives.
