Product Rule

3 Key excerpts on "Product Rule"

• 

  eBook - ePub

  Mathematical Methods for Finance

  Tools for Asset and Risk Management

  • Sergio M. Focardi, Frank J. Fabozzi, Turan G. Bali(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)
• A sequence or a function tends to a finite limit if there is a number to which the sequence or the function can get arbitrarily close; a sequence or a function tends to infinity if it can exceed any given quantity. Starting from these simple concepts, rules for computing limits can be established and limits computed.
• A derivative of a function is the limit of its incremental ratio when the interval tends to zero. Derivatives represent the rate of change of quantities.
• The derivative of the product of a constant and a function is the product of the constant and the derivative of the function.
• The derivative of a sum of functions is the sum of derivatives and called termwise differentiation.
• The derivative of a product of functions is the derivative of the first function times the second plus the first function times the derivative of the second and is called the Product Rule.
• The derivative of a function of functions is the product of the outer function with respect to the inner function times the derivative of the inner function and is called the chain rule.
• A derivative of order n of a function is defined as the function that results from applying the operation of derivation n times.
• A function that is differentiable to any order at a given point a can be represented as a series of the powers of (x – a ) times the n- th derivative at a times the reciprocal of n! ; this expansion is called a Taylor series expansion.
• A Taylor series expansion series is used to approximate the value of a function.
• Taylor series truncated to the first or second terms are called first- and second-order approximations, respectively.
• A special case of the Taylor series is a Maclaurin series. The McLaurin series is the Taylor series computed around x = 0.
• Differentiation can be extended to functions of more than one variable.
• A function of n variables has n first derivatives, n- square second derivatives, and so on.
1. The rate of change should not be confused with the return on an asset, which is the asset’s percentage price change.
2. If the short-term rate is variable:
3. When interest rates are deterministic but time-dependent, the derivative dV /di is computed as follows. Assume that interest rates experience a parallel shift i (t ) + x and compute the derivative with respect to x evaluated at x =