Question

What is the geometrical interpretation of a derivative? What is the difference between partial derivatives and ordinary derivatives?

Short answer

Answer: The main difference between partial derivatives and ordinary derivatives is that partial derivatives are used for multivariable functions, examining the rate of change with respect to one variable while holding all others constant, while ordinary derivatives are used for single-variable functions, examining the rate of change with respect to the only variable in the function.

Step-by-step solution

Geometrical Interpretation of a Derivative

A derivative of a function is a measure of how much the function's output (or dependent variable) changes in response to a small change in its input (or independent variable). Geometrically, the derivative of a function at a given point corresponds to the slope of the tangent line to the function's graph at that point. In simpler terms, the derivative represents the rate of change (or the slope) of the function at a specific point.

Partial Derivatives

Partial derivatives are used when dealing with multivariable functions (functions with more than one independent variable). A partial derivative is the rate of change of the function with respect to one of its variables while keeping the other variables constant. It is denoted by the symbol ∂, for example, ∂f/∂x, which represents the partial derivative of the function f(x, y) with respect to x, keeping y constant.

Ordinary Derivatives

Ordinary derivatives, on the other hand, are used when dealing with single-variable functions (functions with only one independent variable). An ordinary derivative represents the rate of change of the function with respect to its only variable and is denoted by d, for example, df/dx, which represents the derivative of the function f(x) with respect to x.

Difference between Partial Derivatives and Ordinary Derivatives

The primary difference between partial derivatives and ordinary derivatives lies in the context in which they are used. Partial derivatives apply to multivariable functions and examine the rate of change with respect to one variable while holding all others constant. Ordinary derivatives apply to single-variable functions and examine the rate of change with respect to the only variable in the function. Though the concept and interpretation of both types of derivatives are similar, they are used in different contexts and their notation is different, as well.