Calculate the derivative of the following functions.
y = sec(3x+1)

Verified step by step guidance

1. Step 1: Identify the outer function and the inner function. Here, the outer function is \( \sec(u) \) and the inner function is \( u = 3x + 1 \).
2. Step 2: Recall the derivative of the secant function. The derivative of \( \sec(u) \) with respect to \( u \) is \( \sec(u) \tan(u) \).
3. Step 3: Apply the chain rule. The chain rule states that the derivative of a composite function \( y = f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \).
4. Step 4: Differentiate the inner function \( u = 3x + 1 \) with respect to \( x \). The derivative \( \frac{du}{dx} \) is 3.
5. Step 5: Combine the results using the chain rule. The derivative of \( y = \sec(3x + 1) \) is \( \sec(3x + 1) \tan(3x + 1) \cdot 3 \).

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 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 it can be calculated using various rules, such as the power rule, product rule, quotient 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 trigonometric functions with linear transformations.

Secant Function

The secant function, denoted as sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). It is important in calculus, especially when dealing with derivatives of trigonometric functions. The derivative of sec(x) is sec(x)tan(x), and understanding this relationship is crucial for differentiating functions that involve secant.