Find the derivative of the following function:

To take this derivative we want to use logarithmic differentiation. To that end we take the derivative of both sides,

Therefore, taking the derivative of both sides, we have

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Stumbling Robot

A Fraction of a Dot
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Tag: Products

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Find the derivative of a product of terms *(x-a*_{i})^{bi}

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Conjecture an inequality for products

** Claim: ** In addition to we also need for each .

* Proof. * The statement is certainly true for the case (since by assumption). Assume then that it is true for some . Then,
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Use an induction and properties of the product to prove an identity

* Proof. * For the case on the left we have,
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Prove the telescoping property of products

* Proof. * For the case we have,
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Prove the multiplicative property of products by induction

* Proof. * The proof is by induction. For , we have
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Give an inductive definition of the product symbol

* Definition. *
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Establish a formula for (1-1/4)(1-1/9)…(1-1/n^2)

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Establish a formula for the product (1-1/2)(1-1/3)…(1 – 1/n)

Find the derivative of the following function:

To take this derivative we want to use logarithmic differentiation. To that end we take the derivative of both sides,

Therefore, taking the derivative of both sides, we have

If for , what conditions are needed for the inequality

to hold?

by the inductive hypothesis. But then, since ,

Thus, the statement is true for ; and hence, for all

Prove the identity:

for .

While, on the right we have,

So, the identity holds in the case . Assume then that it holds for some . Then we have,

Hence, the statement is true for , and so, for all

Use induction to prove that if for all , then

This is the telescoping property for products.

Thus, the property holds for the case . Assume then that it holds for some . Then,

Hence the property is true for ; and thus, for all

Prove the multiplicative property of the product, i.e.,

Thus, the multiplicative property holds for the case . Assume then that it holds for some . Then,

Thus, the case is true; therefore, the property holds for all

Give an inductive definition of the symbol

** Claim: **

* Proof. * If , we have on the left, and on the right we have . Thus, the formula is true for .

Assume then that the formula is true for some . Then,

Thus, if the formula is true for then it is true for . Since we have established that it is true for , we then have that it is true for all

** Update: ** From a request in the comments, we’ll add in a way to arrive at the formula (without just guessing).

First, we write,

Then, we consider the product

Where in the last line we cancelled terms again. The only things we are left with are the in the numerator and the 2 and in the denominator. Of course, this is pretty much a proof that the formula is correct without using induction, but it doesn’t rely on us guessing the formula correctly.

As noted in the comments, often it is easier to guess the correct formula and use induction to prove the formula is correct than to derive the formula directly.

** Claim: **

* Proof. * If on the left we have and on the right we have . Thus, the formula is true for the case .

Assume then that the formula is true for some . So,

Thus, if the formula is true for then it is true for . Since we have established that it is true for , we have that is true for all