Calculate the derivative of the following functions.
y = tan(xe^x)

Verified step by step guidance

1. Step 1: Identify the outer function and the inner function. Here, the outer function is \( \tan(u) \) and the inner function is \( u = xe^x \).
2. Step 2: Apply the chain rule for differentiation, which states that \( \frac{dy}{dx} = \frac{d}{du}[\tan(u)] \cdot \frac{du}{dx} \).
3. Step 3: Differentiate the outer function \( \tan(u) \) with respect to \( u \). The derivative of \( \tan(u) \) is \( \sec^2(u) \).
4. Step 4: Differentiate the inner function \( u = xe^x \) with respect to \( x \). Use the product rule, which states that \( \frac{d}{dx}[uv] = u'v + uv' \), where \( u = x \) and \( v = e^x \).
5. Step 5: Combine the results from Step 3 and Step 4 to find \( \frac{dy}{dx} = \sec^2(xe^x) \cdot (e^x + xe^x) \).

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative is often denoted as f'(x) or dy/dx and can be calculated using various rules, such as the power rule, product rule, and chain rule.

Product Rule

The product rule is a formula used to find the derivative of the product of two functions. If u(x) and v(x) are two differentiable functions, the product rule states that the derivative of their product is given by u'v + uv'. This rule is essential when differentiating functions that are products of simpler functions, such as in the case of y = tan(xe^x).

Chain Rule

The chain rule is a technique for differentiating composite functions. If a function y is defined as a composition of two functions, such as y = f(g(x)), the chain rule states that the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is particularly useful when dealing with functions like tan(u), where u itself is a function of x, such as xe^x in this case.