Relationship between differentials and derivatives of functions

Differential calculus - Wikipedia

relationship between differentials and derivatives of functions

Suppose that x and y are real numbers and that y is a function of x, that is, This relationship can be written as y = f(x). Geometrically, the derivative of f at the point x = a is the slope of the tangent line to the function. The big idea of differential calculus is the concept of the derivative, which essentially The graphical relationship between a function & its derivative (part 1). for differentiating functions. We will discuss the Product Rule and the Quotient Rule allowing us to differentiate functions that, up to this point.

Calculus is fundamentally different from mathematics which not only uses the ideas from geometry, arithmetic, and algebra, but also deals with change and motion. The calculus as a tool defines the derivative of a function as the limit of a particular kind.

relationship between differentials and derivatives of functions

The concept of derivative of a function distinguishes calculus from other branches of mathematics. Differential is a subfield of calculus that refers to infinitesimal difference in some varying quantity and is one of the two fundamental divisions of calculus. The other branch is called integral calculus.

Differential is one of the fundamentals divisions of calculus, along with integral calculus. It is a subfield of calculus that deals with infinitesimal change in some varying quantity. The world we live in is full of interrelated quantities that change periodically. For example, the area of a circular body which changes as the radius changes or a projectile which changes with the velocity.

Difference Between Differential and Derivative | Difference Between | Differential vs Derivative

These changing entities, in mathematical terms, are called as variables and the rate of change of one variable with respect to another is a derivative. And the equation which represents the relationship between these variables is called a differential equation.

Differential equations are equations that contain unknown functions and some of their derivatives. The concept of derivative of a function is one of the most powerful concepts in mathematics.

The derivative of a function is usually a new function which is called as the derivative function or the rate function. Example 2 Differentiate each of the following functions.

Here is the work for this function. In fact, it is easier. There is a point to doing it here rather than first.

relationship between differentials and derivatives of functions

In this case there are two ways to do compute this derivative. There is an easy way and a hard way and in this case the hard way is the quotient rule.

So, what was so hard about it? However, having said that, a common mistake here is to do the derivative of the numerator a constant incorrectly. For some reason many people will give the derivative of the numerator in these kinds of problems as a 1 instead of 0! Also, there is some simplification that needs to be done in these kinds of problems if you do the quotient rule.

So the slope looks like it is-- I'm just trying to eyeball it-- so the slope is a constant positive this entire time. We have a line with a constant positive slope. So it might look something like this. And let me make it clear what interval I am talking about. I want these things to match up. So let me do my best.

So this matches up to that. This matches up over here. And we just said we have a constant positive slope. So let's say it looks something like that over this interval.

Derivatives: how to find derivatives | Calculus | Khan Academy

And then we look at this point right over here. So right at this point, our slope is going to be undefined. There's no way that you could find the slope over-- or this point of discontinuity. But then when we go over here, even though the value of our function has gone down, we still have a constant positive slope. In fact, the slope of this line looks identical to the slope of this line.

Taking derivatives

Let me do that in a different color. The slope of this line looks identical. So we're going to continue at that same slope. It was undefined at that point, but we're going to continue at that same slope. And once again, it's undefined here at this point of discontinuity. So the slope will look something like that. And then we go up here.