Sketching the Derivative: Step-by-Step Guide
What is Sketching the Derivative? (Simple Explanation)
Sketching the derivative means drawing the graph of a function’s derivative based on its original graph. The derivative of a function represents its rate of change or slope at any given point. By observing the shape and slope of the original function, you can create a new graph that shows how the slope changes over time.

Why It’s Important:
Sketching the derivative helps students visualize complex functions and understand how different sections of the graph relate to the function’s rate of change. It is especially useful in calculus when learning about slope, concavity, and the overall behavior of functions.
Step-by-Step Example:
- Given a Function’s Graph: Start with the original graph, which could be curved or straight.

- Find Key Points:
- Positive Slopes: Where the graph is increasing, the derivative is positive.
- Negative Slopes: Where the graph is decreasing, the derivative is negative.
- Flat Points: At peaks, valleys, or flat sections, the derivative is zero (i.e., crosses the x-axis).

- Draw the Derivative: Create a graph that shows these changes in slope. Positive slopes on the original graph correspond to positive values on the derivative graph, and negative slopes correspond to negative values. Where the slope is zero, the derivative graph touches the x-axis.

How to Sketch the Derivative of a Function?
Let’s break it down with an example:
- Start with a function’s graph: Imagine we have a graph of a curve that rises, flattens, then falls.
- Identify the slopes: In areas where the original graph is increasing (sloping upward), the derivative is positive. Where the graph is decreasing (sloping downward), the derivative is negative. At any peak or valley where the slope is flat (horizontal), the derivative is zero.
- Draw the derivative: Based on the slope changes, you would draw a graph that shows positive values where the original graph slopes up, negative values where it slopes down, and crosses the x-axis at points where the slope is zero.
Example: If you have a graph of a hill, the slope is steep and positive as you climb the hill, then flattens at the top (slope = 0), and becomes negative as you go down. The derivative graph would start high, hit zero at the top, and go below the axis as the hill declines.
Why It’s Useful: Sketching the derivative helps you visualize how a function changes and can provide insights into the behavior of more complex systems.
Understanding the Concept of a Derivative

The derivative of a function represents the slope of the tangent line to the curve at any given point. It shows how the function’s value changes with respect to changes in the input value.
Steps to Sketch the Derivative of a Function
- Identify Key Points on the Original Function:
- Look for points where the function has a horizontal tangent (slope = 0). These are points where the derivative will cross the x-axis.
- Determine intervals where the function is increasing or decreasing.
- Determine the Slope Behavior:
- Identify points where the function is concave up or concave down. In concave up regions, the derivative is increasing, while in concave down regions, the derivative is decreasing.
- Plot the Derivative:
- Sketch the behavior of the derivative based on the slope information. Points where the original function is increasing will have a positive derivative, and points where it is decreasing will have a negative derivative.

Example:
For f(x)=x3−3×2+2xf(x) = x^3 – 3x^2 + 2xf(x)=x3−3×2+2x:
- Identify Key Points:
- Critical points where the slope is 0: solve f′(x)=3×2−6x+2=0f'(x) = 3x^2 – 6x + 2 = 0f′(x)=3×2−6x+2=0.
- Determine Intervals:
- Analyze the intervals to understand where fff is increasing or decreasing.
- Sketch the Derivative:
- Use the identified points and intervals to sketch the derivative function.
Common Mistakes to Avoid
- Misidentifying the critical points where the slope is zero.
- Confusing the original function’s shape with its derivative.
- Not considering the concavity of the function when sketching the derivative.
Practical Applications
- Understanding motion: velocity and acceleration as derivatives of position.
- Analyzing rates of change in economics and biology.
Conclusion: Sketching the derivative of a function is a fundamental skill in calculus that involves understanding the slope behavior of the original function. By following the steps outlined, you can accurately visualize and sketch the derivative.