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Magazine Chapter 3: Problem 31
Find the derivatives of the function. $$y=x^{3} e^{x}$$
Short Answer
Expert verified
The derivative is \( y' = e^x (x^3 + 3x^2) \).
Step by step solution
01
Identify the Function
The function we need to differentiate is given by \( y = x^3 e^x \). This function is a product of two functions: \( u(x) = x^3 \) and \( v(x) = e^x \).
02
Recall the Product Rule for Derivatives
The derivative of a product of two functions \( u(x) \) and \( v(x) \) is given by the product rule: \( (uv)' = u'v + uv' \). We will use this rule to find the derivative of \( y = x^3 e^x \).
03
Find the Derivative of \( u(x) = x^3 \)
Differentiate \( u(x) = x^3 \) with respect to \( x \). The derivative \( u'(x) = 3x^2 \).
04
Find the Derivative of \( v(x) = e^x \)
Differentiate \( v(x) = e^x \) with respect to \( x \). The derivative \( v'(x) = e^x \) since \( e^x \) is its own derivative.
05
Apply the Product Rule
Substitute \( u'(x) = 3x^2 \), \( u(x) = x^3 \), \( v(x) = e^x \), and \( v'(x) = e^x \) into the product rule formula: \( (uv)' = u'v + uv' \). This gives:\[ y' = 3x^2 e^x + x^3 e^x \]
06
Simplify the Expression
Factor out \( e^x \):\[ y' = e^x (3x^2 + x^3) \]Combine the terms inside the parenthesis as:\[ y' = e^x (x^3 + 3x^2) \]This is the 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.
Product Rule
The product rule is a fundamental concept in calculus that allows us to find the derivative of a product of two functions. When you have a function like \( y = x^3 e^x \), which is a product of \( u(x) = x^3 \) and \( v(x) = e^x \), you can't simply take the derivatives of each part and multiply them together. Instead, you apply the product rule.
- The product rule is mathematically expressed as \((uv)' = u'v + uv'\).
- This means that to differentiate \( y = x^3 e^x \), you must compute \( u'(x) \) and \( v'(x) \) first.
- Then, substitute these derivatives into the product rule formula.
Differentiation
Differentiation is the process of finding the derivative of a function. A derivative measures how a function changes as its input changes. It is a powerful tool in calculus, especially when understanding rates of change and slopes of curves.
- Differentiation involves applying rules like the power rule, product rule, and chain rule to functions.
- For the function \( y = x^3 e^x \), differentiation allows us to find \( y' \), which represents the rate of change of \( y \) with respect to \( x \).
- We found \( y' = e^x (x^3 + 3x^2) \) using differentiation techniques.
Exponential Function
The exponential function is an important type of mathematical function often encountered in problems involving growth and decay. The function \( e^x \) is a prime example of an exponential function. One of its key properties is that it is its own derivative.
- The formula \( v(x) = e^x \) implies that the derivative \( v'(x) = e^x \).
- This unique property makes computations involving \( e^x \) particularly straightforward when differentiating.
- In our problem, this characteristic means that you don't need to do further manipulations to find its derivative.
