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) = (x+4)/(4-x)

Verified step by step guidance

1. First, identify the function given: \( f(x) = \frac{x+4}{4-x} \). We need to find the derivative \( f'(x) \) to express the differential \( dy \).
2. Apply the quotient rule for differentiation, which states that if \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). Here, \( u(x) = x+4 \) and \( v(x) = 4-x \).
3. Calculate the derivatives: \( u'(x) = 1 \) and \( v'(x) = -1 \).
4. Substitute these into the quotient rule formula: \( f'(x) = \frac{(1)(4-x) - (x+4)(-1)}{(4-x)^2} \). Simplify the expression to find \( f'(x) \).
5. Finally, express the differential relationship as \( dy = f'(x)dx \), where \( f'(x) \) is the simplified derivative from the previous step.

Key Concepts

Differentials

Differentials represent the relationship between small 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 the function. This concept 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. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In the context of the given function, f'(x) will provide the slope of the tangent line at any point on the curve, which is essential for calculating dy.

Function Composition

Function composition involves combining two functions to create a new function. In this case, understanding how the function f(x) = (x+4)/(4-x) behaves is crucial for finding its derivative. Analyzing the function's structure helps in applying the quotient rule for differentiation, which is necessary for determining f'(x) and subsequently expressing dy in terms of dx.