Dy/dx: must dx be the independent variable?

In summary, the conversation discusses the relationship between x and y in the equations y=sin(x) and x=sin(y). It is noted that the derivative of y=sin(x) can also be found using implicit differentiation and that the inverse function, x=arcsin(y), can also be used to find the derivative. It is also mentioned that there is no distinction between "independent" and "dependent" variables in mathematics and that the Pythagorean Theorem can be used to find the derivative of x=sin(y). The conversation concludes with the understanding that the derivative formula for arcsin(x) is 1/\sqrt{1- x^2}.
  • #1
Eus
94
0
Hi Ho!

If [tex]y=\sin x[/tex], [tex]\frac{dy}{dx}=\frac{d(\sin x)}{dx}=\cos x[/tex].

If [tex]x=\sin y[/tex], [tex]y=\arcsin x[/tex], and therefore [tex]\frac{dy}{dx}=\frac{d(\arcsin x)}{dx}=\frac{1}{\sqrt{1-x^2}}[/tex].

But, if [tex]x=\sin y[/tex], can [tex]\frac{dy}{dx}[/tex] be done as [tex]\frac{dy}{dx}=\frac{dy}{d(\sin y)}=\frac{1}{\frac{d(\sin y)}{dy}}}=\frac{1}{\cos y}=\sec y[/tex]?

If it cannot, is it because the definition of [tex]\frac{dy}{dx}[/tex] that says that [tex]\frac{dy}{dx}=\lim_{\Delta x\rightarrow 0}{\frac{f(x+\Delta x)-f(x)}{\Delta x}}[/tex] requires that [tex]dx[/tex] must be the independent variable and [tex]dy[/tex] must be the dependent variable? Or, is there any other reason?

Thank you very much.


Eus
 
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  • #2
Actually, in most mathematics there is little or no distinction between "independent" and "dependent" variables. All we really need is that there be a "relation" between x and y. If y= sin(x), as long as we stay on an interval in which sine is "one-to-one", we can define the inverse function x= arcsin(y). Another way to find the derivative of x= sin(y), with respect to x, is to use "implicit differentiation": take the derivative of both sides with respect to x. 1= cos(y)dy/dx (by the chain rule) so dy/dx= 1/cos(y)= sec(y). If it bothers you that the right side is a function of y instead of x, you can change that by using trig identities, or even simpler thinking of y as an angle in a right triangle. If x= sin(y), then we can take x as the "opposite side" of a right triangle with hypotenuse 1. y is the angle opposite side x of course, so sin(y)= opposite side/near side= x/1= x. By the Pythagorean Theorem, then, the length of the "near side" is [itex]\sqrt{1- x^2}[/itex]. Since secant is defined as "hypotenuse over near side", sec(y)= [itex]1/\sqr{1- x^2}[/itex]. That is, if x= sin(y), dx/dy= [itex]1/\sqrt{1- x^2}[/itex] which is precisely the derivative formula for "arcsin(x)".
 

Related to Dy/dx: must dx be the independent variable?

1. What is the meaning of "dx" in Dy/dx?

The "dx" in Dy/dx represents the change in the independent variable, which is typically denoted as "x". It is used in calculus to represent an infinitesimal change in the independent variable.

2. Why is "dx" considered the independent variable in Dy/dx?

In the notation Dy/dx, "dy" represents the dependent variable and "dx" represents the independent variable. This is because, in calculus, the derivative is the rate of change of the dependent variable with respect to the independent variable.

3. Can "dx" be replaced with another variable in Dy/dx?

No, "dx" cannot be replaced with another variable. It is a standard notation used in calculus to represent the independent variable and its infinitesimal change. Replacing "dx" with another variable would change the meaning of the notation.

4. What happens if "dx" is not the independent variable in Dy/dx?

If "dx" is not the independent variable in Dy/dx, then the notation becomes ambiguous. The derivative represents the rate of change of the dependent variable with respect to the independent variable. If "dx" is not the independent variable, it is unclear what the derivative is being taken with respect to.

5. Is "dx" always used as the independent variable in calculus?

Yes, "dx" is always used as the independent variable in calculus. It is a standard notation that is universally accepted and understood in the field of mathematics. Other notations, such as "dt" for time, may be used in specific contexts, but "dx" is the most commonly used variable for the independent variable in calculus.

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