Limit Definition for the Derivative. This is the slope of the tangent line, or derivative at point P. This gives us the instantaneous rate of change of y with respect to x. Notice that when the slope of the parabola is negative, the function of the derivative is below zero, and when the slope of the parabola is positive, so is the function of the derivative. When the parabola dips and the slope changes from negative to positive, the function of the derivative goes from negative to positive.
Different notations for the derivative. Finding the derivative for some functions is harder than others, and can be a tedious process when using the slope formula. Luckily, there is an easier way of obtaining the derivative of polynomials without using limits. Newton and Leibniz discovered an easy way to find the derivative of harder functions that only takes a few steps. Let's look at an example:. The first step to finding the derivative is to take any exponent in the function and bring it down, multiplying it times the coefficient.
We bring the 2 down from the top and multiply it by the 2 in front of the x. Then, we reduce the exponent by 1. For the second term, the exponent is assumed to be 1, so we bring it down and multiply it by the coefficient in front of the x. Then, we reduce the exponent by 1, making it 0. The third term is eliminated because it does not have an x, which means it is a constant.
The reason for this is because the number 3 can be written as 3x 0 , and when the 0 comes down the whole term becomes 0.
The Geometrical Concept of the Derivative
Now we are left with our simplified derivative:. Notice that the derivative is linear and the original function is quadratic. The derivative will always be one degree less than the original function. Here is a general rule for taking the derivative of all terms of a polynomial where c is a constant:. This is commonly called the Power Rule see proof of power rule.
Now, you must be careful when finding the derivative, because not every function has one. Most functions are differentiable, which means that a derivative exists at every point on the function. Some functions, however, are not completely differentiable. The limit as h approaches 0 from the left is different than when h approaches 0 from the right.
This is equivalent to saying the derivative or slope on the left is -1, whereas the derivative of the right side is 1. What is the slope where they meet at the origin? Looking at the graph, we can see that at the origin there is not a definite slope because there are multiple tangents, so there is not a derivative at that point.
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We must note that in order for a function to be differentiable, it must be continuous. Earlier, we found the slope of the tangent line at a point using the limit definition of a derivative. Let's do an example finding the tangent line at a given point using the power rule for polynomials.
Now we can use point slope form to find the equation of the tangent line. Isaac Newton focused on the physical concept of differentiation as it applied to mechanics and instantaneous rate of change. As it relates to mechanics, the rate of change is defined as velocity, or speed, when we are talking about distance over a period of time. Just like the geometrical approach, visualize that you are traveling from point A to point B.
We use the formula for the slope to find the average velocity:. Now, if we want to find the instantaneous velocity, we want the change in time to get smaller and smaller. We introduce the concept of a limit as the change in time approaches 0. We end up with. Notice that this is the exact same as the geometric definition of the derivative, but with different variables. The physical definition is based off of the geometric definition, and all of the rules of derivatives apply to both. While you can find velocity by taking the derivative, you can also find the acceleration by taking the second derivative, i.
The acceleration is found by taking the derivative of the velocity function, or the second derivative of the position. Let's analyze the graph from a physical perspective.
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The black curve is the object's position. Notice that when the curve has a hump, the velocity function hits 0. Picture an object going a certain distance in a straight line and then coming back -- the object cannot turn around without the velocity resting at 0. This is the same for the acceleration as it relates to the velocity function. To see if a critical point is a local maximum or minimum you can use the second derivative test. This video shows you how! This video will prepare you for learning implicit differentiation.
A technique to find the formula for the derivative without actually determining explicit formulas for the functions first. What is the definition of the derivative? How can you differentiate standard functions and other functions? This video gives the summary of week 5 of the edX course: pre-university Calculus Calcx. Study Menu sluiten. Research Menu openen. Cooperation Menu openen.
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