Name: graphing the derivative function
Uploaded: 2026-05-14T13:32:49.795Z
Duration: 1 min 26 s
Channel: JoVE

The derivative measures how the dependent variable changes with respect to the independent variable.Graphically, the derivative is the limit of the average rate of change as the interval approaches zero.In mathematical terms, the derivative function, f&#x2032;(x), assigns a slope to each point on the graph of f(x) &#x2014;showing how the output changes in response to changes in the input.When calculated at every point in a function&#x2019;s domain, it produces a new function&#x2014;the derivative function.For instance, at point A, the graph of f(x) is decreasing, and the tangent line has a negative slope. So, the derivative f&#x27;(x) takes on a negative value at this point, corresponding to the point A on the graph of f(x).At point B, the tangent is horizontal or the slope is zero, which means the derivative function is zero. At point C, the slope is positive, shown by a positive value of the derivative function.As a result, the derivative function shows how rapidly the original function is changing at each point.One example of a derivative is found in motion, where the derivative of a car&#x2019;s velocity with respect to time gives an acceleration function.

the derivative as a function

A derivative quantifies how a function changes in response to variations in its input. It provides a localized rate of change, representing the slope of the tangent line to the function at any given point. When this process is applied systematically across the entire domain of the function, it yields a new function&#x2014;the derivative function&#x2014;which encodes the rate of change at every point. This concept is central to calculus and essential for understanding the behavior of dynamic systems in both natural and engineered contexts.<h3>The Derivative Function</h3>Given a differentiable function f(x), its derivative function f&#x2032;(x) assigns to each value of x the instantaneous rate at which f(x) is changing. Formally, it is defined as the limit of the average rate of change as the interval becomes infinitesimally small:<figure class="image image-style-align-center" style="display:table;margin-left:auto;margin-right:auto;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/40113/40113_Equation_1_editor_40113.svg" alt="40113_Equation_1.svg" /></figure>This expression, if the limit exists, gives the slope of the tangent to the curve at x. Thus, f&#x2032;(x) captures the sensitivity of the output to small changes in the input. While the original function describes a quantity&#x2019;s value across its domain, the derivative function reveals how that value evolves locally.<h3>Interpretation and Application</h3>The derivative function plays a critical role in analyzing dynamic systems. In applied contexts, it models quantities such as velocity, growth rate, and marginal cost. For example, in motion analysis, the position function describes location over time, and its derivative&#x2014;the velocity function&#x2014;indicates speed and direction at each moment.Graphically, the derivative function reflects the geometry of the original function&#x2019;s graph: where f(x) is increasing, f&#x2032;(x)&gt;0; where f(x) is decreasing, f&#x2032;(x)&lt;0; and where f&#x2032;(x)=0, the function has a horizontal tangent, potentially indicating a local extremum. Through this lens, the derivative function serves as a powerful analytical tool in both theoretical and practical investigations.

The Derivative as a Function

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JoVE Core

Calculus

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Derivatives

KEY TERMS AND CONCEPTS

graphing the derivative function

The derivative function describes how a function changes at each point. It gives the instantaneous rate of change, which is the slope of the tangent line to the graph at a specific x-value. When this idea is applied across the whole domain, it creates a new function that records the rate of change everywhere.

For a differentiable function f(x), the derivative function is written as f′(x). It assigns each input x the instantaneous rate at which f(x) is changing. Formally, it comes from the limit of the average rate of change as the interval becomes very small. If that limit exists, it gives the slope of the tangent line at x.

This means the derivative function shows how sensitive the output is to small changes in the input. The original function gives the value of a quantity over its domain. The derivative function shows how that value changes locally.

The derivative function is important for studying dynamic systems. In real situations, it can model velocity, growth rate, and marginal cost. In motion, a position function gives location over time, and its derivative, the velocity function, gives speed and direction at each moment.

The graph of the derivative also helps explain the shape of the original function. Where f(x) is increasing, f′(x) is positive. Where f(x) is decreasing, f′(x) is negative. Where f′(x) equals 0, the graph has a horizontal tangent, which may mark a local extremum. Taken together, these ideas make the derivative function a powerful tool for analyzing change.

Graphing the Derivative Function

tangent line

Secant Slope to Tangent Line

Differential calculus helps describe how a quantity changes at a specific point. A moving car on a winding road gives a clear example. The car’s path is a continuous curve, and the direction it travels at any instant is shown by the tangent to that curve. A secant line, by contrast, crosses the curve at two points. It shows how the car’s position changes over an interval, so it describes average behavior rather than an instant. The slope of the secant line between two points on a function f(x)...

derivatives

Square Root Derivatives and Tangent Slopes

Derivatives describe instantaneous rate of change, which is a key idea in both mathematics and physics. They are used to show how a moving object changes position over time. A derivative at a point is also the slope of the tangent line to a curve at that point. It tells how much the output of a function changes for a tiny change in input. For the square root function, finding this rate of change starts with the limit of the difference quotient. The algebra is harder here because the numerator...

rates of change

Average and Instantaneous Change in Functions

Rates of change describe how one variable changes in response to another. In math, this idea helps model dynamic systems in physics, biology, economics, and engineering. It also helps students study real-world functions that change over time or across different inputs. The average rate of change measures the overall change in a function over an interval [x1, x2]. It is found by dividing the change in output by the change in input. On a graph, this value is the slope of the secant line, which...

The derivative function describes how a function changes at each point. It gives the instantaneous rate of change, which is the slope of the tangent line to the graph at a specific x-value. When this idea is applied across the whole domain, it creates a new function that records the rate of change everywhere. For a differentiable function f(x), the derivative function is written as f′(x). It assigns each input x the instantaneous rate at which f(x) is changing. Formally, it comes from the...

higher derivatives

Concavity and Motion with Higher Derivatives

Higher derivatives show how a function changes from one step to the next. In calculus, the first derivative gives the slope of a graph, and the second derivative shows how that slope is changing. These ideas help explain both the shape of a curve and the motion of an object. For the example function f(x) = x^3 - x, the first derivative is f'(x) = 3x^2 - 1. The second derivative is f''(x) = 6x. In this case, the second derivative is a linear function, so the slope changes at a steady rate. <br...

rate of change: problem solving

Brook Trout Growth vs Water Temperature

Brook trout growth changes with water temperature. The data show how trout weight shifts over a 24-day period at several temperatures. At cooler water temperatures, such as 15.5 degrees Celsius, the trout gain a lot of weight. As the temperature rises, the amount of weight gained gets smaller. At the warmest temperature tested, 24.4 degrees Celsius, the trout have a net loss in weight. This pattern suggests that brook trout do better in cooler water. Warmer water may create metabolic stress or...

key terms and concepts

Learn how the derivative function shows slope, rate of change, and motion. See how positive, negative, and zero values connect to a graph.

Graphing the Derivative Function | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Graphing the Derivative Function

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