Difference between differentiating a function and an equation

[autodidude]

What is the difference? I always see differentiate a function but never an equation, a lot of exercises have y=blahblah which is an equation. Does it just mean that when you're asked to differentiate the equation (without using implicit), that it is satisfies the conditions for a function?

 

[chiro]

What is the difference? I always see differentiate a function but never an equation, a lot of exercises have y=blahblah which is an equation. Does it just mean that when you're asked to differentiate the equation (without using implicit), that it is satisfies the conditions for a function?

The difference is that the function case is a special case of the general case of implicit differentiation: your function is explicit with respect to the rest of the variables.

It's basically akin to the difference of say d/dx(f) instead of say d/dx(f*x) where first is df/dx and the second is x*df/dx + f.

Again its best if you think of a function as just another variable (this is it all it is) and that instead of the variable f being inter-twined where it can't be easily algebraically separated, it is explicit which means you can put f on one side and all the other variables on the other.

 

[HallsofIvy]

The reason you only see "differentiate a function" is that is the way differentiation is defined. You don't differentiate an equation, you differentiate the functions on the two sides of the equals sign.

 

[autodidude]

^ Ooh, ok, thank you!

 

[voko]

Only identities can be differentiated meaningfully. Equations are usually valid only for a select number of unknown values, and the "differentiated equation" may have a completely different set of roots. For example: $$3x = x^2 + 2$$has roots $$x = 1, 2$$However, differentiating, we have$$3 = 2x$$ where the root is $$x = \frac 3 2$$ This is why it rarely makes sense to differentiate equations (unless we are talking about functional equations).