Calculate the derivative of the following functions.
y = sin(sin(e^x))

Verified step by step guidance

1. Step 1: Identify the outermost function and the inner functions. Here, the outermost function is \( \sin(u) \), where \( u = \sin(v) \) and \( v = e^x \).
2. Step 2: Apply the chain rule. The chain rule states that the derivative of a composite function \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \).
3. Step 3: Differentiate the outer function \( \sin(u) \) with respect to \( u \). The derivative is \( \cos(u) \).
4. Step 4: Differentiate the middle function \( \sin(v) \) with respect to \( v \). The derivative is \( \cos(v) \).
5. Step 5: Differentiate the innermost function \( e^x \) with respect to \( x \). The derivative is \( e^x \).

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 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.

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 dealing with nested functions, such as in the given problem where sine functions are composed with an exponential function.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental functions in mathematics that relate angles to ratios of sides in right triangles. In calculus, these functions have specific derivatives: the derivative of sin(x) is cos(x). Understanding the behavior and derivatives of these functions is crucial when calculating derivatives of more complex expressions involving trigonometric functions.