# Derivatives of Basic Functions

## The Derivative of x

Derivative formula given for functions of the form f(x) = xn, derived using the binomial theorem.

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## Another Proof That d(xn)/dx = nx(n-1)

Proof by induction of derivative formula for xn.

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## The Derivative of un

Derivative formula for un, proven by induction.

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## The Derivative of xa, a a Fraction

Derivative formula found for functions of the form f(x) = xa, where a is a fraction.

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## Rules for Exponentials and Logarithms

Algebraic rules for exponentials and logarithms are reviewed.

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## The Derivative of ax

Derivation, leading to the definition of e and the natural logarithm.

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## The Derivative of log_a(x) and the Value of e

Derivation using the chain rule. Derivative of ln(x) also given and used to find the numeric value of e.

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## Logarithmic Differentiation

Finding the derivative of a product of functions using logarithms to convert into a sum of functions. Includes worked example.

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## Trigonometric Functions

Angles and continuous functions of them are described abstractly, with mention of the specific functions sin, cos, tan, sec, csc, and cot.

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Four questions which involve calculating derivatives of trigonometric functions.

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## Trigonometric Identities

Angle addition formulas and other trigonometric identities involving sin and cos.

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## Derivatives of sin(x) and cos(x)

Derivation using trig identities and difference quotients.

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## Derivatives of Other Trigonometric Functions

Derivative of tan(x) is derived from the quotient rule and the derivatives of sin(x) and cos(x). Derivatives for sec(x), csc(x), and cot(x) are also stated.

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## The Inverse Trigonometric Functions

Brief definitions of the inverse trigonometric functions sin-1(x), cos-1(x), and tan-1(x)

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## Derivatives of the Inverse Trigonometric Functions

Formulas for the derivatives of the inverse trigonometric functions, as well as the equation sin-1(x) + cos-1(x) = pi/2.

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## The Exponential Function

Definition, including the properties of the function and its derivatives, as well as a graph of the function.

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## Properties of Trigonometric Functions

List of important properties, as well as the derivatives of sine and cosine and a power series representation of sine and cosine.

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## Philosophic Implications

Deriving further rules for derivatives, including the product rule and the rule for functions of the form xn.

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## Derivatives of the Basic Functions

Derivatives for the identity, exponential, and sine functions.

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## Tangent Line to the Graph of an Exponential Function

Finding the equation for the tangent line to an exponential function through a point not on the graph of the function.

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## Derivatives of Exponential and Trigonometric Functions

Taking the first and second derivatives of a function involving an exponential and a cosine.

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## First, Second, and Third Derivatives

Finding the derivatives of an exponential function.

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## Computing Derivatives

Finding the derivatives of an exponential function.

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## Derivatives of Trigonometric Functions

Finding the derivatives of two trigonometric functions.

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## Derivative of an Inverse Function

Evaluating the derivative of the inverse of an exponential function.

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## Evaluating Derivatives and Limits

Four-part question involving the evaluation of three derivatives and a limit.

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## The Inverse Sine Function

Sketching the graph of the inverse sine function and finding its derivative.

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## Evaluating Derivatives

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Two questions finding the derivatives of functions.

Finding the derivatives of four functions.

Three derivatives to be evaluated using a variety of techniques.

## Trigonometric Formulas and Identities

A list of trigonometric and inverse trigonometric identities and formulas involving integrals and derivatives.

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## Differentiation Formulas: Polynomials, Products, Quotients

Five questions which involve taking derivatives and antiderivatives of polynomials, finding the points on a graph which have horizontal tangent lines, and finding derivatives of rational functions.

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## Higher Derivatives

Five questions which involve finding second, third, or nth derivatives of functions.

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## Exponentials and Logarithms: Calculus

Five questions which involve finding second, third, or nth derivatives of functions.

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## Differentials and Indefinite Integration

Three questions which involve evaluating five differentials and twenty indefinite integrals using a range of techniques.

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## Inverse Trigonometric and Hyperbolic Functions

Six questions which involve evaluating integrals and derivatives of these functions, as well as graphing them and finding tangent lines or average values.

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