Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = sin² x

Verified step by step guidance

1. First, identify the function given: \( f(x) = \sin^2 x \). This is the function whose differential we need to find.
2. To find \( dy \), we need to determine \( f'(x) \), the derivative of \( f(x) \). Use the chain rule to differentiate \( \sin^2 x \). Let \( u = \sin x \), then \( f(x) = u^2 \).
3. Differentiate \( u^2 \) with respect to \( u \) to get \( 2u \). Then differentiate \( \sin x \) with respect to \( x \) to get \( \cos x \).
4. Apply the chain rule: \( f'(x) = 2u \cdot \cos x = 2 \sin x \cdot \cos x \).
5. Express the differential relationship: \( dy = f'(x)dx = 2 \sin x \cos x \cdot dx \). This shows how a small change in \( x \) results in a change in \( y \).

Key Concepts

Differentials

Differentials represent the infinitesimal changes in variables. In calculus, if y is a function of x, the differential dy is defined as the product of the derivative f'(x) and the differential dx, which represents a small change in x. This relationship helps in approximating how a small change in x affects the change in y.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is denoted as f'(x) and is calculated as the limit of the average rate of change of the function as the interval approaches zero. In the context of the given function, f(x) = sin² x, the derivative will provide the slope of the tangent line at any point on the curve.

Chain Rule

The chain rule is a fundamental theorem 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 is particularly useful for functions like f(x) = sin² x, where the inner function is sin x.