# Why Does the Expression -dy/dx/dx/dy Calculate the Slope at a Point?

• ktpr2
In summary, the code in the first post will find the slope of a function at a given point, and the code in this post will find the derivative of the function at a given point.
ktpr2
I recently wrote a program to check implicit differentiation and the key bit I saw in other programs was the logic amounting to:

$$-\frac{ \frac{dy}{dx}} {\frac{dx}{dy}}$$

when given a valid (x,y) point on the curve. Can someone explain why the above generates the slope at point (x,y)?

ktpr2 said:
I recently wrote a program to check implicit differentiation and the key bit I saw in other programs was the logic amounting to:

$$-\frac{ \frac{dy}{dx}} {\frac{dx}{dy}}$$

when given a valid (x,y) point on the curve. Can someone explain why the above generates the slope at point (x,y)?

Are you sure you don't mean something like:
$$\frac{ \frac{dy}{dt}} {\frac{dx}{dt}}$$

This is just the operation of the chain rule:
dy/dt = (dy/dx) (dx/dt)

so dy/dx = (dy/dt)/(dx/dt)

Well that's the code snippet I used. To take the derivative of x in terms of y seems kinda heady. The actual code was:

(-der1(eqn,x))/(der1(eqn,y)), which means, according to my ti-86 manual, returns the value of the first derivative ... with respect to variable for the current variable value. This is provided, as a point on the curve.

So I suppose I'm actually just taking the derivative of the expression in terms of a constant. I'm not quite sure what that would mean or if it's nothing more than normal division?

EDIT - Actually, it's probably the derivative of the expression with the value for x or y plugged in. I should probably research dx/dy because that's what's going on here.

Last edited:
In the way you've written it,i.e.that ration,i hope u see that it makes no sense whatsoever.

Daniel.

Well, that's the program code and it works just fine. I do believe my teacher said you can take dy/dx, just that we won't be doing it anytime soon. I'll talk to him about it.

Why are you trying to find dx/dy? I'm confused on what the purpose of your program is. It seems there is some practical purpose, so I just don't see where dx/dy fits in. The code in the first post would find $$\frac{d^2y}{dx^2}$$ if I'm not mistaken.

Jameson

It sounds like you mean:

$$\frac{d y}{d x} = - \frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}$$

Is this right? If so, this is the equation for the slope of the contour of constant f passing through some point (x,y) in a 2D contour plot of f(x,y). This formula works because if you change x by some amount, you change f by df/dx*dx. Then to get back on the contour, you must change y by enough to get the opposite change, df/dy*dy=-df/dx*dx. This gives you your formula.

statusX- yeah that seems like what's going on.

Jameson -THe program merely checks your answer for implict differentation; you find a point on the curve via TI 8X's Solver function, and run it and it'll give you the slope and ask you to type in your answer. It compares and tells you if you're right or not. (http://www.ticalc.org/pub/86/basic/math/imp.zip )

Last edited by a moderator:
ktpr,

What are you entering into the calculator to get dx/dy? I'm guessing it's something like "der(y,y)" where you have defined that first "y" to be your function. But that would give you dy/dy which is 1 and would explain why your ratio comes out to be simply dy/dx.

Am I close?

## 1. Why is the notation -dy/dx/dx/dy used in calculus?

The notation -dy/dx/dx/dy is used in calculus to represent the second derivative of a function. It is derived from the notation for the first derivative, dy/dx, which represents the rate of change of y with respect to x. The second derivative is the rate of change of the rate of change, and so the notation -dy/dx/dx/dy is used.

## 2. How does -dy/dx/dx/dy relate to the concavity of a function?

The second derivative, -dy/dx/dx/dy, is used to determine the concavity of a function. If the second derivative is positive, the function is concave up, meaning it is curving upwards. If the second derivative is negative, the function is concave down, meaning it is curving downwards. The second derivative also helps to identify inflection points in a function where the concavity changes.

## 3. Can -dy/dx/dx/dy be simplified?

Yes, the notation -dy/dx/dx/dy can be simplified to d2y/dx2. This notation is used more commonly in mathematics and is equivalent to -dy/dx/dx/dy. It represents the second derivative of a function, which is the rate of change of the rate of change of the function.

## 4. How is -dy/dx/dx/dy used in optimization problems?

In optimization problems, the second derivative, -dy/dx/dx/dy, is used to determine the maximum or minimum points of a function. The maximum point occurs when the second derivative changes from positive to negative, and the minimum point occurs when the second derivative changes from negative to positive. These points can be identified using the second derivative test.

## 5. What is the physical interpretation of -dy/dx/dx/dy?

The physical interpretation of -dy/dx/dx/dy is the acceleration or deceleration of a changing rate. In other words, it represents how fast the rate of change is changing. This can be applied to real-world situations, such as the acceleration of a moving object or the growth rate of a population.

• Introductory Physics Homework Help
Replies
1
Views
764
• Differential Equations
Replies
5
Views
540
• Introductory Physics Homework Help
Replies
2
Views
842
• Science and Math Textbooks
Replies
5
Views
2K
• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
936
• Introductory Physics Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
863
• Calculus and Beyond Homework Help
Replies
10
Views
2K
• Calculus
Replies
10
Views
2K