How do you graph the derivative of a function when you are given the graph of the function?

Answer 1
You do this by looking at how the slope of the lines tangent to the graph change as the value of #x# (the independent variable) changes.
This works, because the derivative gives us a formula (a function) for finding the slopes of tangent lines for various values of #x#.
Consider the following graph of a function below. Notice that as we look from left to right, the tangents on the left have large positive slope. The slope decreases (the tangents are more nearly horizontal) as we pass x=0 (the #y#-axis) and get close to the high point around x=0.5. The #y# value here is a bit more than 2 and it is called a local or a relative maximum value). Continuing our rightward journey, the slopes of the tangent lines become negative and decrease to about x=1.5 after which point the tangents once again get flatter (closer to horizontal). We arrive a a local minimum value when we reach 2.3 or so and continue into a part of the graph where the tangent lines have positive slope. graph{x^3-4x^2+2x+2 [-3.19, 7.91, -2.93, 2.62]} Here it the graph of the derivative of the function above: graph{3x^2-8x+2 [-3.416, 6.45, -3.848, 1.087]}
Notice that the local extreme values for the function occur at the same #x# values that make the derivative #0#. (The value of the function is #0# at the #x#-intercept of the graph.
Should you wish to try this with a graphing utility and other functions: try: #f(x)=x^5-x^4-3x^3+x^2-x# and #f'(x)=5x^4-4x^3-9x^2+2x-1#.
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Answer 2

To graph the derivative of a function when you are given the graph of the function, follow these steps:

  1. Identify the critical points of the function on the given graph. These points occur where the function has maximum, minimum, or points of inflection.
  2. Determine the intervals where the function is increasing or decreasing based on the slope of the original function's graph.
  3. Use the information from step 2 to determine the sign of the derivative in each interval. If the function is increasing, the derivative is positive; if the function is decreasing, the derivative is negative.
  4. Identify any points of discontinuity or sharp changes in the graph of the original function. These may indicate points where the derivative is undefined or has a vertical tangent.
  5. Plot the critical points, intervals of increase and decrease, and any points of discontinuity on the graph of the derivative.
  6. Sketch the graph of the derivative, ensuring that it reflects the sign changes and behavior of the original function.

By following these steps, you can graph the derivative of a function using the given graph of the function.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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