What is the correct second derivative for implicit differentiation of r^2 = x^2?

In summary: But now you have found your mistake and it should be clear. Good job!In summary, the conversation revolved around finding the second derivative of the equation r^2 = x^2 and the correct result being (r^2 - x^2)/r^3. The individual made a mistake in their working by dividing instead of subtracting, but was able to correct it with help.
  • #1
K41
94
1
I have an equation:

r^2 = x^2

So I found out dr/dx = x/r.

But when I try to find the second derivative, I get d2r/dx2 = -x^2/r^3 when the text says it should be (r^2 - x^2)/r^3.

Can anyone help? My working out:

r^2 - x^2 = 0
r^2 = x^2.
Assume r is a function of x.
rr' = x (first derivative found correctly)
rr'' + r'(x/r) = 1 (apply chain rule and sub in answer for first derivative)
rr'' + x^2/r^2 = 1 (sub in first derivative)

So where have I gone wrong?
 
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  • #2
djpailo said:
I have an equation:

r^2 = x^2

So I found out dr/dx = x/r.

But when I try to find the second derivative, I get d2r/dx2 = -x^2/r^3 when the text says it should be (r^2 - x^2)/r^3.

Can anyone help? My working out:

r^2 - x^2 = 0
r^2 = x^2.
Assume r is a function of x.
rr' = x (first derivative found correctly)
rr'' + r'(x/r) = 1 (apply chain rule and sub in answer for first derivative)
rr'' + x^2/r^2 = 1 (sub in first derivative)

So where have I gone wrong?

And then where do you go from that last line?
 
  • #3
Only in the very last steps (after your last line):
rr'' + x2/r2 = 1 ⇔
rr'' = 1 - x2/r2 ⇔
r'' = 1/r - x2/r3 ⇔
r'' = ( r2 - x2 ) / r3
 
  • #4
Haha, you won't believe what I was doing. Instead of subtracting both sides, I was doing a division (for reasons not clear to me or anyone of the known realm)...

GGGAAAAAHHHH

Thanks!
 
  • #5
I believe you. You are not the only one...
 
  • Like
Likes DrewD and K41

Related to What is the correct second derivative for implicit differentiation of r^2 = x^2?

What is implicit differentiation?

Implicit differentiation is a mathematical technique used to find the derivative of a function that is not explicitly expressed in terms of one of its variables. It is particularly useful for finding the derivatives of equations that cannot be easily solved for one variable, such as equations involving both x and y.

How does implicit differentiation differ from explicit differentiation?

In explicit differentiation, the function is expressed in terms of one variable, making it easy to find the derivative. In implicit differentiation, the function is not explicitly expressed in terms of one variable, so the chain rule and implicit differentiation rules must be used to find the derivative.

Why is implicit differentiation useful?

Implicit differentiation allows us to find the derivatives of equations that are difficult or impossible to solve for one variable. This technique is particularly helpful in solving problems in physics, engineering, and economics, where equations often involve multiple variables.

What is the process of implicit differentiation?

The process of implicit differentiation involves using the chain rule and implicit differentiation rules to find the derivative of an equation with respect to a specific variable. First, the equation is rewritten in the form of f(x,y)=0. Then, the derivative of both sides of the equation is taken with respect to the desired variable, treating the other variable as a function of x. Finally, the resulting equation is solved for the desired derivative.

What are some common mistakes to avoid in implicit differentiation?

Some common mistakes to avoid in implicit differentiation include not applying the chain rule correctly, not differentiating all terms in the equation, and not properly treating the variables as functions of x. It is also important to carefully keep track of the variables and their derivatives throughout the process.

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