**ABSTRACT**

Calculus is an important subject since it exists in most of university courses such as economy, engineering, statistics, science, and all mathematical courses like numerical analysis, statistic, differential equation, and operation research. Derivatives is one of the important concepts of calculus which is a precondition topic for most of mathematics courses and other courses in different fields of studies. Different differentiation techniques can be designed to help students in learning of derivatives based on mathematical thinking and generalization strategies. Solving for the nth derivative of some functions is a tedious task especially if successive differentiations will be executed. The main aim of this study was to derive formulas for determining the nth derivative of some standard functions in selected forms. Also, it aimed to evaluate the Taylor Series Expansion and Maclaurin Series of select functions using the derived formulas for the nth derivative. In this study, basic research was employed to derived formulas for the nth derivative of some functions. Expository method was used in developing the algorithms. Proofs through mathematical induction were presented to guarantee the generalization of the assertion.

* Keywords*: derivatives, differential calculus, nth derivative, maclaurin series, power functions, taylor series expansion

**INTRODUCTION**

Calculus is an important subject since it exists in most of university courses such as economy, engineering, statistics, science, and all mathematical courses like numerical analysis, statistic, differential equation, and operation research.

Calculus is one of the fundamental and underlying branches of mathematics. It was known before as infinitesimal calculus, which is focused on functions, infinite series and sequences, limits, derivatives, and anti-derivatives (Guce, 2013). For most students in mathematics, science, and engineering, calculus is the entry-point to undergraduate mathematics.

Differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation.

The main aim of this study was to derive formulas for determining the nth derivative of some standard functions in selected forms. It also aimed to evaluate the Taylor Series Expansion and McLaurin Series of select functions using the derived formulas for the nth derivative. The formulas that were generalized were different from existing equations regarding nth derivatives.

**METHODS**

In this study, basic research was employed to derive formulas for the nth derivative of select functions. The manner of deriving formulas employed expository method. First, pertinent data needed in solving the nth derivatives were gathered. The researcher conducted series of trials to discover pattern which may lead to conjectures in certain cases. Afterwards, proofs of the conjectures were illustrated. Mathematical induction was employed to guarantee the generalization of the assertion. In the principle of mathematical induction, the conjecture satisfied two important properties which are the base case and the inductive hypothesis. After deriving the formulas for the nth derivative of select functions, the Taylor Series Expansion and Maclaurin Series of those functions in selected forms were evaluated since it involves nth derivatives.

**RESULTS AND DISCUSSION**

The following formulas for the nth derivative of selected functions were derived. The nth derivative formulas were applied to Maclaurin Series and Taylor Series Expansion.

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