If you want, you could change the number of points used to calculate the slope - it would be fun. So you can see two things. First, the derivative is just the rate the function changes for very tiny time intervals.
Second, this derivative can usually be written as another actual mathematical function. In general, we write the derivative as:.
That's it. Oh, you think it's cheating to use a computer? Ok, I can understand that. But really, it's not cheating. A numerical program takes a derivative by using finite but very tiny time intervals.
In real life, this is always what we are dealing with and science deals with the real world. But how do you get a mathematical function without using a computer?
I'm not going to go over all the details - that's what your math class is for. All I care about as a physics coach is that you understand what a derivative is and how to find it. So, here are some "rules". Product rule. You never have just a plain function. Suppose I have a function g and f both are functions of t. Now I have a position function x t such that:. I can find the derivative of this function by finding the derivative of g t and f t in the following manner.
Power rule. If you have a polynomial, it's pretty easy to find the derivative. Suppose I have a function like this:.
Where n is just a constant. In that case, the derivative of this function will be:. Trig functions. Remember, I am not deriving these.
I am just telling you "the answer" - so here are derivatives of the two most common trig functions. I cheated - I skipped a small step above. In order to really understand the trig derivatives, you also need the chain rule. Chain rule. What if you have a function of a function a composite function? Here is an example. That one might be a little harder to explain - let's just hope you cover that in your math course soon. Remember that the derivative is really just a rate of change.
If we have a curve of a function and we want to find the equation of the tangent to a curve at a given point, then by using the derivative, we can find the slope and equation of the tangent line. A tangent is a line to a curve that will only touch the curve at a single point and its slope is equal to the derivative of the curve at that point.
Similarly, we can find the equation of the normal line to the curve of a function at a point. This normal line will be normal perpendicular to the tangent line. Application of derivatives is also helpful in finding the maxima, minima, and point of inflection of a curve. Maxima and minima are the peaks and valleys of a curve, whereas the point of inflection is the part of the curve where the curve changes its nature from convex to concave or vice versa.
We can find the maxima, minima, and point of inflection by using the first-order derivative test. According to this test, we first find the derivative of the function at a given point and equate it to 0, i.
Now if the function is defined in the given interval, then we check the value of f' x at the points lying to the left of the curve and to the right of the curve and check the nature of the f' x , then we can say, that the given point is maxima or minima based on the below conditions. By using derivatives, we can find out if a function is an increasing or decreasing function. The increasing function is a function that seems to reach the top of the x-y plane whereas the decreasing function seems like reaching the downside corner of the x-y plane.
Let us say we have a function f x which is differentiable within the limits a, b. Then we check any two points on the curve of the function. A few places where we will use the derivative are given below and then explained one by one in the following sections. The most common usage of the application of derivatives is seen in the following areas. Derivatives are also helpful in finding the maxima, minima, and point of inflection of a curve.
Derivatives represent a rate of change. In mathematics, a rate of change can be applied to many circumstances. For instance, acceleration is the rate of change in velocity.
Therefore, a derivative function can be used to determine the acceleration of an object when the velocity over time is given. The second coke you will drink just after will provide you utility but less.
The increase of your utility will be less fast more you consume. This one can be formulated by the second derivative of the utility function as follows ;. Sign up to join this community.
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I think it would be bad pedagogy to include such details when trying to explain the first time to someone who doesn't yet understand calculus. Neal Neal 30k 2 2 gold badges 61 61 silver badges bronze badges.
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