Implicit differentiation question

In summary, the chain rule is needed to differentiate y3. The product rule is used to simplify the derivative of 100x and get the answer 100x.
  • #1
Obsidian
7
0
Find y' = dy/dx for x3 + y3 = 4


Okay, now what's really confusing me is that for the y3 is that you need to use the chain rule for it. When you do, the answer is 3y2(dy/dx). How does that actually work?

And if anyone can give me any good advice on any good guidelines on how to properly implicitly differentiate, it would be most helpful. Thanks. :)
 
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  • #3
My next question is, in my book, they differentiate d/dx(100xy) and turn it into:

100[x{dy/dx) + y]

How did they get to that?
 
  • #4
Product rule. You can't just think of y as a constant when you differentiate implicitly.
 
  • #5
Yeah, I felt it had something to do with the product rule, but why did they take out the 100 like that?
 
  • #6
It's just a constant, so that can be moved out. d/dx[100xy] = 100 d/dx[xy] = 100[x*dy/dx + dx/dx*y]
 
  • #7
But can't you do that only if the 100 is being multiplied in both x and y? I mean, the way it is, wouldn't it only be multiplying to either the X or the y, not twice to both of them?
 
  • #8
You really have to know algebra in order to do calculus! 100xy= (100x)y= x(100y)= 100(xy).
 
  • #9
No, I know that's how it works, but I guess my question is, why isn't the derivative of 100 being taken as well? Why is it being left out?

Sorry for the dumb questions. :/
 
  • #10
Obsidian said:
No, I know that's how it works, but I guess my question is, why isn't the derivative of 100 being taken as well? Why is it being left out?

Sorry for the dumb questions. :/
Ok, take the derivative of this problem. Treat it as if it's a product.

[tex]\frac{d}{dx}(100x)[/tex]

What is your answer?

Now go back to your derivative properties in which, [tex]\frac{d}{dx}cf(x)=cf'(x)[/tex]

Now do you see why it can be left out?
 
  • #11
Ah, of course! Me so dumb. Thanks a lot. :)
 

Related to Implicit differentiation question

1. What is implicit differentiation?

Implicit differentiation is a mathematical technique used to find the derivative of a function that is not given in the form of y=f(x). It is often used when the dependent variable (y) cannot be easily isolated on one side of the equation.

2. How is implicit differentiation different from explicit differentiation?

Explicit differentiation is used to find the derivative of a function that is explicitly written in the form of y=f(x), where the dependent variable (y) is already isolated. Implicit differentiation, on the other hand, is used when the dependent variable (y) cannot be easily isolated and requires the use of the chain rule.

3. When should implicit differentiation be used?

Implicit differentiation should be used when the function is given in an implicit form, meaning the dependent variable (y) is not explicitly written in terms of x. It is also useful when the function is too complex to be solved using traditional methods.

4. What is the process for solving an implicit differentiation question?

The process for implicit differentiation involves taking the derivative of both sides of the equation with respect to x, using the chain rule when necessary. Then, solve for the derivative of y by moving all terms with dy/dx to one side of the equation and all other terms to the other side. Finally, solve for dy/dx to find the derivative of the original function.

5. Can implicit differentiation be used for multivariable functions?

Yes, implicit differentiation can also be used for multivariable functions where there are more than one independent variable. In this case, the partial derivative with respect to the dependent variable can be found by treating all other variables as constants and applying the same process as with a single variable function.

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