# Can you use derivatives and integrals to solve algebraic problems?

• Rib5
In summary, derivatives and integrals can be used to change the form of an equation to make it easier to solve a problem. However, taking the derivative of both sides of an equation does not always result in the same equality holding. This is because the two functions are only conditionally equal, not identically equal. Additionally, the derivative can also be used to solve infinite sums, as shown in the provided example.
Rib5
I've seen derivatives and integrals used before to change the form of an equation to one that is more suitable for solving a problem. Usually the person will just differentiate both sides and the equality holds? Is this always the case?

Because if you take the derivative of both sides of x^2 = 4, you get 2x = 0, which is not right. Can someone explain how this works?

I am not aware that you can do such things, do you have an example of where this was used?

But yes the equality will hold once you do the same thing on both sides of the equation.

What you have found in your example is the value of x for which x^2 - 4 has a horizontal tangent.

What you might be thinking of is that if two functions are identically equal (equal for all values of x), then their deriviatives are equal. In your example, the functions x^2 and 4 (or x^2 - 4 and 0) are not identically equal; they are only conditionally equal.

In my book they do use the derivative to solve an infinite sum.

http://img268.imageshack.us/img268/6097/probf.gif

 Ah ok! Thanks!

Last edited by a moderator:
This summation is identically true for all values of alpha, so the derivatives will also be equal.
$$\sum_{i = m}^n \alpha^i~=~\frac{\alpha^m - \alpha^{n + 1}}{1 - \alpha}$$

## 1. Can derivatives be used to solve algebraic problems?

Yes, derivatives can be used to solve algebraic problems. In fact, derivatives are commonly used to find the rate of change or slope of a function at a specific point, which can be helpful in solving algebraic equations.

## 2. How can integrals be applied to solve algebraic problems?

Integrals can be used to find the area under a curve, which can be helpful in solving algebraic problems involving geometric shapes or finding the total change in a function over a specific interval.

## 3. Can derivatives and integrals be used interchangeably to solve algebraic problems?

No, derivatives and integrals cannot be used interchangeably to solve algebraic problems. While derivatives focus on finding the rate of change or slope of a function, integrals focus on finding the area under a curve. They are two different mathematical techniques that serve different purposes.

## 4. Are there any limitations to using derivatives and integrals to solve algebraic problems?

Yes, there can be limitations to using derivatives and integrals to solve algebraic problems. Some problems may be too complex to solve using these techniques, and in those cases, other methods may be necessary.

## 5. Can derivatives and integrals be used to solve any type of algebraic problem?

No, derivatives and integrals cannot be used to solve all types of algebraic problems. They are most commonly used in problems involving rates of change, optimization, and area under a curve. Other types of problems may require different techniques or approaches.

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