Question

Compute the derivative of the given function. $$f(x)=\left(3 x^{2}+8 x+7\right) e^{x}$$

Short answer

The derivative of the function is \(f'(x) = (3x^2 + 14x + 15)e^x.\)

Step-by-step solution

Recognize the Product Rule

The function given is the product of two functions, \(u(x) = 3x^2 + 8x + 7\) and \(v(x) = e^x\). To find the derivative of a product of two functions, we use the product rule: \[\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x).\]

Differentiate First Function \(u(x)\)

Find the derivative of \(u(x) = 3x^2 + 8x + 7\): \[u'(x) = \frac{d}{dx}(3x^2) + \frac{d}{dx}(8x) + \frac{d}{dx}(7).\] This gives: \[u'(x) = 6x + 8.\]

Differentiate Second Function \(v(x)\)

Find the derivative of \(v(x) = e^x\). The derivative of an exponential function \(e^x\) is \[v'(x) = e^x.\]

Apply the Product Rule

Using the product rule formula from Step 1, substitute the derivatives from Steps 2 and 3:\[\frac{d}{dx}[f(x)] = (6x + 8)e^x + (3x^2 + 8x + 7)e^x.\]

Simplify the Expression

Combine the terms:\[f'(x) = (6x + 8)e^x + (3x^2 + 8x + 7)e^x = (3x^2 + 14x + 15)e^x.\] This simplification arises by combining like terms \(6x + 8x = 14x\) and combining constant terms.

Key concepts

Product Rule

The product rule is a crucial concept when finding the derivative of a function that is the product of two different functions. In simple terms, the product rule allows you to differentiate products of functions. This rule states that if you have a function that can be split into two parts, say \( u(x) \) and \( v(x) \), the derivative of their product is given by \( \frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x) \). Here, \( u'(x) \) represents the derivative of \( u(x) \), and \( v'(x) \) represents the derivative of \( v(x) \).

• Step 1: Differentiate the first function, \( u(x) \).
• Step 2: Differentiate the second function, \( v(x) \).
• Step 3: Apply the product rule formula.

Using the product rule, you can efficiently handle derivatives involving products without needing to resort to lengthy algebraic manipulation.

Exponential Function

Exponential functions are a special type of function where a constant base, such as \( e \), is raised to a variable exponent. In calculus, the exponential function \( e^x \) has the unique property where its derivative remains the same, i.e., \( \frac{d}{dx}(e^x) = e^x \). This feature makes exponential functions extremely useful in growth models, financial calculations, and natural phenomena modeling.To differentiate the exponential function \( e^x \):

• Understand that the derivative of \( e^x \) is \( e^x \) itself.
• Recognize that this property also holds true when \( e^x \) is part of a larger function, making calculations straightforward.

This property simplifies differentiation significantly, ensuring that whenever an exponential function is involved, you only need to replicate the function for its derivative.

Differentiation

Differentiation is the process of finding the derivative of a function. It allows you to determine the rate at which a function is changing at any point. In calculus, differentiation can apply to many types of functions including polynomials, trigonometric, and exponential functions. The resulting derivative tells you how the function's value reacts to small changes in input.Key aspects of differentiation include:

• Rules: There are specific rules, like the product rule and chain rule, that facilitate differentiation.
• Applications: Derivatives are used to calculate rates of change, slopes of curves, and optimize solutions in various fields.
• Notation: The derivative of a function \( f(x) \) is typically denoted as \( f'(x) \) or \( \frac{df}{dx} \).

Differentiation empowers you to analyze functions intensely and solve complex calculus problems by assessing how they change with respect to their variables.