Derivatives in Differential Form


In Exercises 17–28, find dy.


y = cos(x²)

Verified step by step guidance

1. Step 1: Identify the function y = cos(x²). This is a composition of functions, where the outer function is cos(u) and the inner function is u = x².
2. Step 2: Apply the chain rule for differentiation, which states that if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
3. Step 3: Differentiate the outer function with respect to its argument u. The derivative of cos(u) with respect to u is -sin(u).
4. Step 4: Differentiate the inner function u = x² with respect to x. The derivative of x² with respect to x is 2x.
5. Step 5: Combine the results from steps 3 and 4 using the chain rule: dy/dx = -sin(x²) * 2x. This is the derivative of y with respect to x.

Key Concepts

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 dy/dx is found by multiplying the derivative of the outer function f with respect to its inner function g, by the derivative of the inner function g with respect to x. This is essential for differentiating y = cos(x²), where x² is the inner function.

Derivative of Trigonometric Functions

Understanding the derivatives of trigonometric functions is crucial for solving problems involving these functions. The derivative of cos(x) with respect to x is -sin(x). This knowledge is applied when differentiating y = cos(x²), where the derivative of the outer function cos is needed to apply the chain rule effectively.

Differential Notation

Differential notation involves expressing the derivative of a function in terms of differentials, such as dy and dx. It provides a way to represent the rate of change of y with respect to x. In the context of the problem, finding dy involves using differential notation to express the derivative of y = cos(x²) in terms of dy and dx, which is useful for understanding changes in y as x varies.