An equality about derivative of a polynomial?

If it is a mistake, it may have been left as an exercise to give students practice in finding and correcting errors. In summary, the textbook presents the equality $$\left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n = 2nx\left(x^2-1\right)^{n-1}$$ and leaves its proof as an exercise. However, it may be a mistake and can potentially be corrected through simple differentiation.
  • #1
td21
Gold Member
177
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Why is $$
\left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n = 2nx\left(x^2-1\right)^{n-1}
$$? This is in a textbook and says that its proof is left as an exercise. It seems to be a difficult equality.

I believe this should just be $$
\left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n = 2nx\left(x^2-1\right)^{n}
$$ by simple differentiation.

Am I wrong or not?
 
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  • #2
td21 said:
Why is $$
\left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n = 2nx\left(x^2-1\right)^{n-1}
$$? This is in a textbook and says that its proof is left as an exercise. It seems to be a difficult equality.

I believe this should just be $$
\left(x^2-1\right)\frac{d}{dx}\left(x^2-1\right)^n = 2nx\left(x^2-1\right)^{n}
$$ by simple differentiation.

Am I wrong or not?
What you have seems fine to me, but I can't say that the book is wrong without actually seeing what is in the book.
 

Related to An equality about derivative of a polynomial?

1. What is the definition of a derivative of a polynomial?

A derivative of a polynomial is the slope of a tangent line to the polynomial's graph at a specific point. It represents the instantaneous rate of change of the polynomial at that point.

2. How do you find the derivative of a polynomial?

To find the derivative of a polynomial, you can use the power rule, which states that the derivative of a term with the form ax^n is nax^(n-1). You can also use the product rule, quotient rule, or chain rule depending on the complexity of the polynomial.

3. What is the relationship between the degree of a polynomial and its derivative?

The degree of a polynomial is equal to the highest power of x in the polynomial. The degree of the derivative of a polynomial is one less than the degree of the original polynomial. In other words, the derivative decreases the degree of the polynomial by one.

4. Can you give an example of finding the derivative of a polynomial?

Sure, let's find the derivative of the polynomial 3x^2 + 2x + 5. Using the power rule, we get 6x + 2. This means that at any point on the graph of the original polynomial, the slope of the tangent line will be 6x + 2.

5. Why is the derivative of a constant value equal to 0?

The derivative of a constant value is equal to 0 because a constant value does not change. The derivative represents the rate of change, so if there is no change, the derivative is 0. In other words, the slope of a horizontal line is always 0.

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