Given a function f that is differentiable on its domain, write and explain the relationship between the differentials dx and dy.

Verified step by step guidance

1. Start by understanding that the differential of a function is a concept used to approximate changes in the function's value. For a differentiable function f(x), the differential dy represents the change in the function's output, while dx represents the change in the input.
2. Recall the definition of the derivative: if f is differentiable at a point x, then the derivative f'(x) is the limit of the ratio of the change in the function to the change in the input as the change in the input approaches zero. Mathematically, this is expressed as: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>&#x2202;</mo><mi>f</mi></mrow><mrow><mo>&#x2202;</mo><mi>x</mi></mrow></mfrac></math>.
3. The relationship between the differentials dx and dy can be expressed using the derivative. If y = f(x), then the differential dy is given by: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>dy</mi><mo>=</mo><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mi>dx</mi></math>. This equation shows that dy is the product of the derivative of f at x and the differential dx.
4. Understand that the differential dy provides an approximation of the change in the function's value for a small change in x. This approximation becomes more accurate as dx approaches zero.
5. Finally, remember that differentials are useful in various applications, such as linear approximations and error estimations, where they help in understanding how small changes in input affect the output of a function.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. It provides a way to understand how a function behaves locally, indicating how small changes in the input (x) affect changes in the output (f(x)).

Differentials

Differentials are infinitesimally small changes in variables, denoted as dx and dy. In the context of a function f(x), dy represents the change in the function's value resulting from a small change dx in the input variable x. The relationship dy = f'(x)dx illustrates how the differential of the function is proportional to the differential of the input, scaled by the derivative.

Chain Rule

The Chain Rule is a key principle in calculus that allows us 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 rule is essential for understanding how changes in one variable affect another through a chain of relationships.