Proving the Product Rule for Differentiating x^n-1

• Perzik
In summary, if you have a function h(x) that is the result of two other functions f(x) and g(x) and if you want to find the derivative of h(x), you use the product rule. First, you need to find the derivative of f(x) and then you use the product rule to find the derivative of g(x).
Perzik

Homework Statement

Use the product rule to show that dx^n-1/dx = (n-1)x^n-2

Homework Equations

The general idea is..

If: h(x) = f(x)g(x)
Then: dh(x)/dx = f(x)dg(x)/dx + g(x)df(x)/d(x)

The Attempt at a Solution

It seems like a simple solution but everytime I attempt solving it I get confused and end up with a totally wrong answer

Define f(x) and g(x) and everything will be easy.

Hint: $$a^n\cdot a^m = a^{n+m}$$

that's the problem I'm getting..I'm not sure where to start to determine f(x) and g(x).

I don't understand this question. What's with the n-1? I'm assuming you don't know the derivative of xn, or at least you're not allowed to use it here, since otherwise you could just replace n with n-1 and get the answer immediately. I guess they want you to derive from scratch what the derivative of xn-1 is. Note (as above) this is equivalent to deriving what the derivative of xn is, which is much less silly. Do you know how to use induction, and can you think of how to apply it here?

StatusX said:
I don't understand this question. What's with the n-1? I'm assuming you don't know the derivative of xn, or at least you're not allowed to use it here, since otherwise you could just replace n with n-1 and get the answer immediately. I guess they want you to derive from scratch what the derivative of xn-1 is. Note (as above) this is equivalent to deriving what the derivative of xn is, which is much less silly. Do you know how to use induction, and can you think of how to apply it here?

Hm, since there's the product rule 'involved', I'd assume one can use the knowledge of what the derivative of x^n equals, but Perzik should know best what the problem asks for.

The point is to use "proof by induction".

If n= 1, what is $dx/dx$? Does that match the formula? What is (1)x0?

Now assume that, for some k,$dx^k/dx= k x^{k-1}$. Write $x^{k+1}$ as $x(x^k)$ and use the product rule.

1. What is the product rule for differentiating x^n-1?

The product rule for differentiating x^n-1 states that the derivative of the product of two functions, f(x) and g(x), is equal to the first function (f(x)) times the derivative of the second function (g'(x)) plus the second function (g(x)) times the derivative of the first function (f'(x)). In mathematical notation, it can be written as (f(x)g(x))' = f'(x)g(x) + f(x)g'(x).

2. Why is it important to prove the product rule for differentiating x^n-1?

Proving the product rule for differentiating x^n-1 is important because it is a fundamental rule in calculus that allows us to find the derivatives of products of functions, which is a common occurrence in mathematical and scientific equations. Understanding and being able to apply the product rule is essential for solving more complex problems in calculus and other branches of mathematics.

3. How do you prove the product rule for differentiating x^n-1?

The product rule for differentiating x^n-1 can be proven using the limit definition of a derivative and the properties of limits. By expanding the product (f(x)g(x))' and simplifying, we can show that it is equal to f'(x)g(x) + f(x)g'(x), thus proving the product rule.

4. Can the product rule be applied to more than two functions?

Yes, the product rule can be applied to any number of functions. For example, if we have three functions f(x), g(x), and h(x), the product rule states that (f(x)g(x)h(x))' = f'(x)g(x)h(x) + f(x)g'(x)h(x) + f(x)g(x)h'(x). This pattern can continue for any number of functions.

5. How can I use the product rule to differentiate a polynomial function?

To differentiate a polynomial function using the product rule, you would first break down the function into simpler terms, such as monomials or binomials. Then, you can use the product rule to find the derivative of each term and combine them using the properties of derivatives. For example, if you have a polynomial function f(x) = x^2 + 3x, you can differentiate it as f'(x) = (x^2)' + (3x)' = 2x + 3.

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