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) = 1/x³

Verified step by step guidance

1. First, identify the function given: \( f(x) = \frac{1}{x^3} \). This is a rational function where the variable x is in the denominator.
2. To find the differential \( dy \), we need to compute the derivative of \( f(x) \) with respect to x, denoted as \( f'(x) \). Use the power rule for derivatives, which states that \( \frac{d}{dx} x^n = nx^{n-1} \).
3. Rewrite \( f(x) = \frac{1}{x^3} \) as \( f(x) = x^{-3} \) to apply the power rule. The derivative \( f'(x) \) is then \( -3x^{-4} \).
4. Express the relationship between the small change in x, \( dx \), and the corresponding change in y, \( dy \), using the formula \( dy = f'(x)dx \). Substitute \( f'(x) \) into this formula: \( dy = -3x^{-4}dx \).
5. The expression \( dy = -3x^{-4}dx \) represents how a small change in x, \( dx \), results in a change in y, \( dy \), for the function \( f(x) = \frac{1}{x^3} \). This is the differential form of the relationship between x and 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 dy = f'(x)dx, where f'(x) is the derivative of f with respect to x. This relationship allows us to approximate how a small change in x (denoted as dx) affects the change in y (denoted as dy).

Derivatives

The derivative of a function measures the rate at which the function's value changes as its input changes. For the function f(x) = 1/x³, the derivative f'(x) can be calculated using the power rule. Understanding how to compute derivatives is essential for expressing the relationship between changes in x and y in differential form.

Chain Rule

The chain rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a variable y depends on u, which in turn depends on x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This concept is crucial when dealing with functions that are not directly in terms of x.