Derivatives in Differential Form


In Exercises 17–28, find dy.


y = sec(x² − 1)

Verified step by step guidance

1. Step 1: Identify the function y = sec(x² − 1). The goal is to find the derivative dy/dx.
2. Step 2: Recognize that y = sec(u) where u = x² − 1. This requires using the chain rule for differentiation.
3. Step 3: Differentiate the outer function sec(u) with respect to u. The derivative of sec(u) is sec(u)tan(u).
4. Step 4: Differentiate the inner function u = x² − 1 with respect to x. The derivative of x² − 1 is 2x.
5. Step 5: Apply the chain rule: dy/dx = (d/dx)[sec(u)] = sec(u)tan(u) * du/dx = sec(x² − 1)tan(x² − 1) * 2x.

Key Concepts

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In this problem, the chain rule helps differentiate y = sec(x² − 1) by first differentiating sec(u) with respect to u, and then differentiating u = x² − 1 with respect to x.

Derivative of Secant Function

The derivative of the secant function, sec(x), is sec(x)tan(x). This derivative is crucial when differentiating y = sec(x² − 1) because it allows us to find the rate of change of the secant function with respect to its argument. Applying this derivative in conjunction with the chain rule helps determine dy/dx for the given function.

Differential Notation

Differential notation involves expressing the derivative in terms of differentials, such as dy and dx. It provides a way to represent the infinitesimal change in y with respect to an infinitesimal change in x. In this problem, finding dy involves using the derivative dy/dx obtained from applying the chain rule and the derivative of the secant function, and then expressing it in differential form as dy = (dy/dx)dx.