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

Verified step by step guidance

1. Step 1: Identify the function to differentiate. The function given is \( y = \csc(e^x) \).
2. Step 2: Recall the derivative of the cosecant function. The derivative of \( \csc(u) \) with respect to \( u \) is \( -\csc(u)\cot(u) \).
3. Step 3: Apply the chain rule. Since \( u = e^x \), differentiate \( u \) with respect to \( x \), which is \( \frac{du}{dx} = e^x \).
4. Step 4: Combine the results using the chain rule. The derivative of \( y = \csc(e^x) \) is \( \frac{dy}{dx} = -\csc(e^x)\cot(e^x) \cdot e^x \).
5. Step 5: Simplify the expression if possible. The derivative is \( \frac{dy}{dx} = -e^x \csc(e^x)\cot(e^x) \).

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 of the function at any given point. The derivative is often denoted as f'(x) or dy/dx, and it can be calculated using various rules such as the power rule, product rule, and chain rule.

Cosecant Function (csc)

The cosecant function, denoted as csc(x), is the reciprocal of the sine function, defined as csc(x) = 1/sin(x). In the context of derivatives, understanding the properties and behavior of trigonometric functions like cosecant is essential, especially when applying differentiation rules. The derivative of csc(x) is -csc(x)cot(x), which is crucial for differentiating functions involving csc.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly important when dealing with functions like y = csc(e^x), where e^x is the inner function.