Derivatives and Integrals of Transcendental Functions Module Overview

Introduction

Finding derivatives and integrals of transcendental functions poses an interesting challenge due to the nonalgebraic characteristic of transcendental functions. To meet this challenge requires an understanding of inverse functions. Further complications arise as none of the six basic trigonometric functions has an inverse function so a redefinition of their domains is necessary. Prepare to investigate interesting relationships between exponential and logarithmic functions and their derivatives and integrals, to discover the origin of the transcendental number e, and to learn a new differentiation technique that uses logarithms in differentiating nonlogarithmic functions.

Essential Questions

What relationship exists between the derivatives of a function and its inverse?
How is implicit differentiation used to find the derivative of an inverse function?
What is a logarithm and how can a natural logarithm be defined in terms of an integral?
Why is the number e so special?
How do the derivative and integral of an exponential function relate?
How are exponential functions used to model real-world situations?
How do you find the derivative of logarithmic functions?
How and when should you use logarithmic differentiation?
How is the definition of the derivative used to find the derivative of the trigonometric functions?
What role do trigonometric functions play in calculus?
Which pairs of inverse trigonometric functions have derivatives that are opposites?
Why are only three rather than six integrals of inverse trigonometric functions given?

Key Terms

The following key terms will help you understand the content in this module.

e - An irrational number whose approximate value is 2.71828182846... and is defined by $LaTeX: e=\lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n$ .

Exponential decay - A function $LaTeX: y=y_0e^{kx}$ where k < 0.

Exponential function - A function of the form $LaTeX: f\left(x\right)=a^x$ , where a is a positive constant.

Exponential growth - A function $LaTeX: y=y_0e^{kx}$ where k > 0.

Inverse function - A function $LaTeX: f^{-1}$ whose domain and range are respectively the range and domain of a given function f, and under which the image, y, of an element, x, is the element of which x was the image under the given function, i.e., $LaTeX: f^{-1}\left(x\right)=y$ if and only if f(y)=x.

Inverse trigonometric functions - Inverse functions for the trigonometric functions with restricted domains (ratios) that make them functions, ranges (angles) that are subsets of the domains of the trigonometric functions, and denoted with a -1 exponent or a prefix of "arc".

Logarithmic differentiation - The process of differentiating after taking logarithms of both sides of an identity.

Logarithmic function - Any function containing the logarithm of a function with base a, a positive constant. $LaTeX: f\left(x\right)=\log_ax$ is the inverse function of the base a exponential function.

Logistic curve - Frequently used to model growth or decay in situations in which saturation (carrying capacity) occurs. It is defined by $LaTeX: y=\frac{a}{1+be^{cx}}+d,a>0\:and\:b>0$

Natural exponential function - A function of the form $LaTeX: f\left(x\right)e^x$ .

Natural logarithmic function - A function of the form f(x) = ln x, where e is the base of the logarithm also defined as $LaTeX: x=\int_1^xt^{-1}dt$

Relative magnitude - The relative size of numbers, distances, or functions based on a comparison or ordering.

Transcendental function - A function not expressible as a finite combination of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and extracting a root

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