Differentiating both sides of an equation

In summary, it is valid to differentiate both sides of an equation when the equation represents a functional relationship between two variables. In the case of your first equation, the relationship holds for all values of r, making it possible to solve for the variable p(r). However, in the second equation, x is not a variable but a constant value, making it impossible to differentiate both sides.
  • #1
awvvu
188
1
When is it valid to differentiate both sides of an equation? I was working on a physics problem and came across this, where I had to solve for p(r).

[tex]q(r) = \int_0^r \rho(s) * 4 \pi s^2 ds = \frac{Q r^6}{R^6}[/tex]

So, differentiating both sides with respect to r and using the Fundamental Theorem:

[tex]\rho(r) * 4 \pi r^2 = \frac{6 Q r^5}{R^6}[/tex]

Solving for p(r), I get the right answer, so obviously this is what they expect me to do. What I'm wondering is why exactly is this valid, when this is not:

[tex]x^2 = x => x = 1[/tex]

Differentiating both sides:

[tex]2 x = 1 => x = \frac{1}{2}[/tex]

I think I came up with the gist of an explanation while typing this post up... but I'd really like a clear and more rigorous way to explain it. x^2 = x is only true for x = 1, so you can't differentiate both sides. But for my first equation, I assume that there is a p(r) that makes both sides of the equation true for all r. So, I can solve for this p(r). Am I right?
 
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  • #2
The problem with your second equation is that it isn't one, at least in the same sense as your first equation. It uniquely determines what x is. Hence, x isn't really a variable - you know what it is - it's 1. It doesn't make sense to differentiate that expression then because x really isn't a variable in it.

In your first expression, you really do have a functional relationship between the integral and the other expression - it holds for any value of r. As a result, r is a variable and the expression can be differentiated on both sides of the equality.

So basically, the explanation you came up with was correct.
 
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  • #3
Okay, thanks.
 

1. What is the purpose of differentiating both sides of an equation?

Differentiating both sides of an equation allows us to find the rate of change or slope of a function, which is useful in many areas of science and mathematics. It also helps us to solve for unknown variables and make predictions about the behavior of a system.

2. How do you differentiate both sides of an equation?

To differentiate both sides of an equation, we use the rules of differentiation, such as the power rule, product rule, and chain rule. These rules allow us to find the derivative of each term in the equation and simplify the expression to solve for the derivative of the entire equation.

3. Can you differentiate both sides of any equation?

No, not all equations can be differentiated. The equation must have a single variable that changes with respect to another variable. If there are multiple variables, we can still differentiate both sides of the equation, but the resulting derivative will be a partial derivative.

4. What happens to the constants when you differentiate both sides of an equation?

In most cases, the constants will disappear when differentiating both sides of an equation. This is because the derivative of a constant is 0. However, if the constant is inside a function, such as f(x) = 2x + 5, the derivative of the function will still include the constant, in this case 2.

5. Why is it important to keep the same operation on both sides of the equation when differentiating?

It is important to keep the same operation on both sides of the equation when differentiating because we are essentially finding the slope of the graph of the equation. If we change the operation on one side, the graph will no longer be accurate and we will not get the correct derivative.

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