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# Find the derivative of a product of terms (x-ai)bi

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 # Conjecture an inequality for products

If for , what conditions are needed for the inequality to hold?

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, by the inductive hypothesis. But then, since , Thus, the statement is true for ; and hence, for all # Use an induction and properties of the product to prove an identity

Prove the identity: for .

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

Use induction to prove that if for all , then This is the telescoping property for products.

Proof. For the case we have, 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 products by induction

Prove the multiplicative property of the product, i.e., Proof. The proof is by induction. For , we have 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 product symbol

Give an inductive definition of the symbol Definition. # Establish a formula for (1-1/4)(1-1/9)…(1-1/n^2)

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.

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

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 