Calculate the derivative of the following functions.
y = tan e^x

Verified step by step guidance

1. Step 1: Identify the function to differentiate. The function given is \( y = \tan(e^x) \).
2. Step 2: Recognize that this is a composition of functions, where \( u = e^x \) and \( y = \tan(u) \). We will use the chain rule to differentiate.
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 \( u = e^x \) with respect to \( x \). The derivative of \( e^x \) is \( e^x \).
5. Step 5: Apply the chain rule: \( \frac{dy}{dx} = \sec^2(e^x) \cdot e^x \). This combines the derivatives from steps 3 and 4.

Key Concepts

Derivative

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

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function y = f(g(x)), the derivative is given by dy/dx = f'(g(x)) * g'(x). This rule is essential when differentiating functions that are nested within each other, such as the function tan(e^x) in the given problem.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental functions in mathematics that relate angles to ratios of sides in right triangles. The derivative of the tangent function, tan(x), is sec²(x). Understanding the derivatives of these functions is crucial for solving problems involving trigonometric expressions, like the one presented in the question.