Question

Compute the derivative of the given function. $$f(x)=x^{2} e^{x} \tan x$$

Short answer

\(f'(x) = 2x e^x \tan x + x^2 e^x \sec^2 x + x^2 e^x \tan x \sec^2 x\).

Step-by-step solution

Identify the Components

The function given is \(f(x) = x^2 e^x \tan x\). It is a product of three functions: \(u = x^2\), \(v = e^x\), and \(w = \tan x\).

Apply the Product Rule

The derivative of a product of three functions \(u\), \(v\), and \(w\) is given by: \[ f'(x) = u'vw + uv'w + uvw' \]We will need to find the derivative of each of \(u\), \(v\), and \(w\).

Differentiate Each Component

1. Differentiate \(u = x^2\):\[ u' = \frac{d}{dx}(x^2) = 2x \]2. Differentiate \(v = e^x\):\[ v' = \frac{d}{dx}(e^x) = e^x \]3. Differentiate \(w = \tan x\):\[ w' = \frac{d}{dx}(\tan x) = \sec^2 x \]

Substitute into the Product Rule Formula

Substitute the derivatives and the original functions into the product rule formula:\[ f'(x) = (2x)(e^x)(\tan x) + (x^2)(e^x)(\sec^2 x) + (x^2)(e^x)(\tan x)' \]

Simplify the Expression

Combine and simplify the terms obtained after substitution:\[ f'(x) = 2x e^x \tan x + x^2 e^x \sec^2 x + x^2 e^x \tan x \sec^2 x \]Notice that we can group similar terms if applicable for simplified expression.

Key concepts

Product Rule

The Product Rule is a very important concept in Calculus, especially when differentiating products of functions. When you have two or more functions multiplied together, like in the expression \( f(x) = u(x) \cdot v(x) \), the product rule helps you differentiate it effectively. The basic formula for two functions is:\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]

• \(u'(x)\) represents the derivative of the first function, \(u(x)\).
• \(v'(x)\) is the derivative of the second function, \(v(x)\).

In the exercise provided, three functions are being multiplied: \(u = x^2\), \(v = e^x\), and \(w = \tan x\). For more than two functions, you use an extended version of the product rule:\[ f'(x) = u'vw + uv'w + uvw' \]This involves taking the derivative of each function, one at a time, while keeping the others unchanged, and then adding these products together.

Derivatives

Derivatives are foundational in calculus, understood as a measure of how a function changes as its input changes. They represent the slope of the tangent line to the function's graph at any given point.
This is extremely useful in various applications, including physics, engineering, and economics.For a function \(f(x)\), its derivative, \(f'(x)\), tells us the rate of change of \(f\) at any point \(x\). Calculating derivatives requires understanding basic rules such as:

• The power rule: For \(x^n\), the derivative is \(nx^{n-1}\).
• The exponential rule: For \(e^x\), the derivative is \(e^x\).
• The trigonometric rule: For \(\tan x\), the derivative is \(\sec^2 x\).

In the context of the original exercise, knowing the derivatives of \( x^2 \), \( e^x \), and \( \tan x \) separately is essential. Each derivative represents how that specific function's value changes as \(x\) changes.

Function Differentiation

Function differentiation involves finding a derivative which effectively provides a function that describes the rate of change of another function. It's about applying our knowledge of derivatives and rules like the product rule to break down complex functions into simpler parts.
To find the derivative of \( f(x) = x^2 e^x \tan x \), you:1. Identify each component of the function that appears as a product.2. Use the product rule to differentiate each part.3. Substitute known derivatives of common functions.Remember, the final step is to combine terms to simplify the expression:

• Combine and reorganize.
• Ensure each term reflects the product rule properly.

This simplification ensures your final derivative \( f'(x) \) is in its most compact and clear form, making it easier to understand and use for interpreting the behavior of \( f(x) \).