How is this Differentiation Trick Manipulated in Calculus?

In summary, the conversation is about the manipulation of equations involving r and x. The first equation is (1/r^2)(d/dr(r^2. dx/dr)) and the second equation is (1/r)(d/dr(r^2.d(xr)/dr)). The speaker is having trouble understanding how to go from the first equation to the second, despite knowing that they are equal.
  • #1
Baggio
211
1
I keep seeing this trick everywhere but I don't see how it is done.

how do we go from

[itex]\frac{1}{r^{2}}\frac{d}{dr}(r^{2}\frac{d \rho}{dr} [\latex]

to

[itex]\frac{1}{r} \frac{d^{2} \rhor}{dr^2{}}[\latex]

Ugggh can't get latex to work anyway it's

(1/r^2)(d/dr(r^2. dx/dr)

how do we go from that to

(1/r)(d/dr(r^2.d(xr)/dr))

I know for sure that they're equal I just don't know how to manipulate it! :(

Thanks
 
Last edited:
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  • #2
Use [ tex ] [ /tex ] (without the spaces).

[tex]\frac{1}{r^{2}}\left(2r\frac{d\rho}{dr}+r^{2}\frac{d^{2}\rho}{dr^{2}}\right)
=\frac{2}{r}\frac{d\rho}{dr}+\frac{d^{2}\rho}{dr} [/tex]

and u can see pretty clearly the 2 things are different.

Daniel.
 
  • #3
Who's "x" and what does he do...?

Daniel.
 
  • #4
No, let's start again

(1/r^2)(d/dr(r^2. dx/dr)

&

(1/r)(d/dr(r^2.d(xr)/dr))

x is a function of r, if you expand both you get the same result, but how can I go directly from the top eq to the bottom
 

1. What is the differentiation trick?

The differentiation trick, also known as the chain rule, is a method used in calculus to find the derivative of a composite function. It allows us to break down a complex function into simpler functions and find the derivative of each part separately.

2. How does the differentiation trick work?

The differentiation trick involves using the chain rule, which states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. This allows us to break down a complex function into simpler functions and find the derivative of each part separately. We then combine these derivatives using the chain rule to find the derivative of the original function.

3. When is the differentiation trick used?

The differentiation trick is used when we need to find the derivative of a composite function, where the input of one function is the output of another. It is commonly used in optimization problems, where we need to find the maximum or minimum value of a function.

4. What are some common mistakes when using the differentiation trick?

One common mistake when using the differentiation trick is forgetting to apply the chain rule and only finding the derivative of the outer function. Another mistake is using the product rule instead of the chain rule when the function is a product of two or more simpler functions.

5. How can the differentiation trick be applied in real-world scenarios?

The differentiation trick has many applications in the real world, such as in physics, engineering, economics, and more. For example, it can be used to find the rate of change in a physical system, the marginal cost in economics, or the optimal design of a structure in engineering. It is a powerful tool that allows us to understand and model complex systems in various fields.

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