Calculate the derivative of the following functions.
y = csc (t² + t)

Verified step by step guidance

1. Step 1: Identify the outer function and the inner function. Here, the outer function is \( \csc(u) \) and the inner function is \( u = t^2 + t \).
2. Step 2: Differentiate the outer function \( \csc(u) \) with respect to \( u \). The derivative of \( \csc(u) \) is \( -\csc(u)\cot(u) \).
3. Step 3: Differentiate the inner function \( u = t^2 + t \) with respect to \( t \). The derivative is \( 2t + 1 \).
4. Step 4: Apply the chain rule. Multiply the derivative of the outer function by the derivative of the inner function: \( \frac{dy}{dt} = -\csc(t^2 + t)\cot(t^2 + t) \cdot (2t + 1) \).
5. Step 5: Simplify the expression if possible to get the final derivative form.

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, representing 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 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). It is important in calculus for understanding the behavior of trigonometric functions and their derivatives. When differentiating functions involving csc, one must apply the chain rule and the derivative of the sine function, which is crucial for accurate calculations.

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 essential when dealing with functions like y = csc(t^2 + t), where the argument of the cosecant function is itself a function of t.