Question

Find \(f^{\prime}(x)\) for each function. $$ f(x)=x^{2} e^{x} $$

Short answer

\( f'(x) = e^x(2x + x^2) \)

Step-by-step solution

Identify the Rule of Differentiation

Since the function is a product of two simpler functions, namely, \(x^2\) and \(e^x\), you need to use the product rule for differentiation. The product rule states that if \( u(x) \) and \( v(x) \) are functions of \( x \), then \( (uv)' = u'v + uv' \).

Assign Functions to Product Rule

Let \( u(x) = x^2 \) and \( v(x) = e^x \). We need to find \( u'(x) \) and \( v'(x) \).

Differentiate \( u(x) \)

Find the derivative of \( u(x) = x^2 \). Using the power rule, \( u'(x) = \frac{d}{dx}(x^2) = 2x \).

Differentiate \( v(x) \)

Find the derivative of \( v(x) = e^x \). The derivative of \( e^x \) is \( e^x \), so \( v'(x) = e^x \).

Apply the Product Rule

Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the product rule formula: \[(u(x)v(x))' = u'(x)v(x) + u(x)v'(x)\]\[f'(x) = (2x)e^x + x^2(e^x)\]

Simplify the Expression

Factor out \( e^x \) from the expression: \[f'(x) = e^x(2x + x^2)\]This is the simplified form of the derivative.

Key concepts

Product Rule

The product rule in calculus is essential for finding the derivative of a product of two functions. If you have two functions, say \( u(x) \) and \( v(x) \), and you want to differentiate their product \( u(x)\cdot v(x) \), the product rule tells you how. The formula is: \[(uv)' = u'v + uv' \] This rule is useful because it gives you a straightforward way to find derivatives without having to manually multiply the functions and then differentiate, which can be cumbersome. Let's break it down: - First, you'll need to find the derivative of the first function \( u(x) \), which we denote as \( u'(x) \).- Next, find the derivative of the second function \( v(x) \), which is \( v'(x) \).- Finally, substitute these derivatives into the formula. Using the product rule is straightforward once you get the hang of it, making it a formidable tool in your calculus toolkit.

Power Rule

The power rule is a basic, yet powerful, tool in calculus for differentiating functions that are powers of \( x \). When you have a function like \( x^n \), where \( n \) is any real number, the power rule allows you to quickly find its derivative. The formula is: \[\frac{d}{dx}(x^n) = nx^{n-1} \] This rule is derived from the limit definition of a derivative, but you don't need to know the limit process to use it effectively. It's a shortcut that saves time and effort. Here’s how it works: - Multiply \( n \) by the coefficient of \( x^n \) which is usually 1 if not stated differently.- Then, decrease the power \( n \) by 1.For example, the derivative of \( x^2 \) is \( 2x \), since applying the power rule results in \( 2x^{2-1} = 2x \). This simplicity is why the power rule is often one of the first techniques taught in calculus and forms the foundation for more complex differentiation rules.

Derivative of Exponential Functions

Exponential functions have the form \( e^x \) where \( e \) is Euler's number, approximately equal to 2.718. The special property of the exponential function \( e^x \) is that its derivative is exactly the same as the original function, \( e^x \). Therefore, we express it as: \[\frac{d}{dx}(e^x) = e^x \] This feature makes differentiating exponential functions particularly straightforward, and allows them to model growth or decay processes precisely. When you're dealing with a function where \( e^x \) is part of an expression, such as when it multiplies another function like \( x^2 \), you'll use a combination of rules, like the product rule, to find the overall derivative. - First, identify the simpler parts of your function.- Differentiate these parts separately—just as you did by finding the derivative of \( e^x \).This dual approach of simplicity and robustness is what makes the concept powerful in calculus applications.