Need a help in solving an equation (probably differentiation

[k.udhay]

I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following equations is a constant.

I tried to get it in y = f(x) format, however because the constants and y are so complexly attached, I failed. It would be of great help, if someone can help me find out an equation that gives me the dy/dx = 0 condition. Thanks.

https://i.stack.imgur.com/cpu8N.jpg

 

[eys_physics]

I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following equations is a constant.

I tried to get it in y = f(x) format, however because the constants and y are so complexly attached, I failed. It would be of great help, if someone can help me find out an equation that gives me the dy/dx = 0 condition. Thanks.
View attachment 215316

https://i.stack.imgur.com/cpu8N.jpg

If $y=f(x)$ you have $x=f^{-1}(y)$, if the inverse $f^{-1}$ exist. As is shown at the Wikipedia page, you then have
$$f'(x)=\frac{1}{x'(y)}$$.

 

[.Scott]

I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following equations is a constant.

I tried to get it in y = f(x) format, however because the constants and y are so complexly attached, I failed. It would be of great help, if someone can help me find out an equation that gives me the dy/dx = 0 condition. Thanks.
View attachment 215316

https://i.stack.imgur.com/cpu8N.jpg

Let's see...
I would go for x=f(y)
So dx/dy = -d(G10+G6)/dy = -tan - d(G10)/dy
Since we are going to be looking for a denominator that goes toh zero, we can drop the tangent term.
So we are looking for 1/(d(G10)/dy)=0.

Attacking the G10 equation:
Let A = D9/H; So G10 = A sqrt( (H/2)^2 - (D12 + Rtip -Sqrt(Rtip^2-y^2) )^2 )
Let B = (H/2)^2 - (D12 + Rtip -Sqrt(Rtip^2-y^2) )^2; So G10 = A sqrt(B)
the derivative of that is:
d(G10)/dy = (A/(2 sqrt(B)) ) (dB/dy) = (A/(2 sqrt(B)) ) ( -(D12+Rtip-sqrt(Rtip^2-y^2))^2 /dy)
Since we are looking for excursions to infinity, the factors "A" and (-(D12+Rtip-sqrt(Rtip^2-y^2))^2) will not assist and can be dropped.
So we are looking for 2sqrt(B) = 0; thus B = 0
So (H/2)^2 = (D12 + Rtip -Sqrt(Rtip^2-y^2) )^2
H = +/-2(D12 + Rtip - sqrt(Rtip^2-y^2) )

Specific information about what's positive and negative helps here.
So I'll leave the rest to you.

BTW: Check everything - I can make mistakes.