Purpose of differentiating both sides

In summary, if y=e^x, rearranging the equation to \ln y=x yields \frac{\delta}{\delta{x}} (ln\ y)=1. Differentiating with respect to x yields \frac{\delta}{\delta{x}} (ln\ y)=1.
  • #1
nobahar
497
2
Hey guys,
I've just recently started calculus, and just trying to get to grips with it.
I'll use an example for my question; if [tex]y=e^x[/tex], then the source suggests rearranging to [tex]\ln y=x[/tex], the differentiating with respect to x yields [tex]\frac{\delta}{\delta{x}} (ln\ y)=1[/tex].

I understand that for my original function I wouldn't be able to find [tex]\delta{y}[/tex], so it's rearranged, but what are the consequences of this? As it appears that the requirement is to DIFFERENTIATE BOTH SIDES. Do I always do this when the equation has been altered? I can't piece togeather why this is happening. Hopefully someone can help.
Thanks.
Nobahar.
 
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  • #2
[tex]y=e^x \Rightarrow lny=x[/tex]


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

[tex]= \frac{dy}{dx}\frac{d}{dy}(lny)=1[/tex]

I didn't fully understand what you were asking so I tried my best to alleviate your confusion.
 
  • #3
rock.freak667 said:
[tex]y=e^x \Rightarrow lny=x[/tex]


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

[tex]= \frac{dy}{dx}\frac{d}{dy}(lny)=1[/tex]

I didn't fully understand what you were asking so I tried my best to alleviate your confusion.

As to the purpose, if you continue with this, since y=e^x then dy/dx=y. So you've shown y*d(ln(y))/dy=1. So d(ln(y))/dy=1/y. You've used the fact you know how to differentiate the exp function to figure out how to differentiate it's inverse function, ln.
 
  • #4
Cheers guys! :smile:
 
  • #5
nobahar said:
Hey guys,
I've just recently started calculus, and just trying to get to grips with it.
I'll use an example for my question; if [tex]y=e^x[/tex], then the source suggests rearranging to [tex]\ln y=x[/tex], the differentiating with respect to x yields [tex]\frac{\delta}{\delta{x}} (ln\ y)=1[/tex].
Other folks showed you what to do, but I don't think they answered your questions, so I'll take a shot at it.
Your goal here was to differentiate e^x, but didn't have a formula to do so.
What you called "rearranging" the equation is to rewrite the original equation, where y is a function of x, as a new equation where x is a different function of y. In fact, the different function is the inverse of the exponential function.

Apparently you already know the derivative of ln(x), so using the chain rule, you can differentiate ln(y) with respect to x (i.e., find d/dx(ln(y)), which as Dick pointed out is 1/y * dy/dx.
nobahar said:
I understand that for my original function I wouldn't be able to find [tex]\delta{y}[/tex], so it's rearranged, but what are the consequences of this? As it appears that the requirement is to DIFFERENTIATE BOTH SIDES. Do I always do this when the equation has been altered?
It doesn't have anything to do with whether an equation has been altered. Whenever you differentiate one side of an equation, you have to differentiate the other side as well. And the differentiation has to be with respect to the same variable. IOW, if you differentiate one side with respect to x, you have to differentiate the other side with respect to x as well.
nobahar said:
I can't piece togeather why this is happening. Hopefully someone can help.
Thanks.
Nobahar.
 
  • #6
That's pretty much answered my questions fully!
Thanks!
 

1. What is the purpose of differentiating both sides?

The purpose of differentiating both sides is to find the rate of change, or the slope, of a function at a specific point. This allows us to analyze and understand how the function behaves and how it changes over time.

2. How does differentiating both sides help in solving equations?

Differentiating both sides can help in solving equations by simplifying them and making them easier to solve. It also allows us to find the critical points of a function, which are the points where the function's derivative is equal to zero.

3. Can differentiating both sides help in visualizing a function?

Yes, differentiating both sides can help in visualizing a function by providing information about its rate of change. This can help us graph the function and understand how it behaves at different points.

4. Is differentiating both sides useful in real-world applications?

Yes, differentiating both sides is useful in real-world applications such as physics, engineering, economics, and many other fields. It allows us to model and analyze real-world phenomena by understanding how they change over time.

5. What are the different methods of differentiating both sides?

There are several methods of differentiating both sides, including the power rule, product rule, quotient rule, and chain rule. These methods involve specific techniques for finding the derivative of different types of functions, such as polynomials, exponential functions, and trigonometric functions.

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