Elementary Calculus: An Infinitesimal Approach - H. Jerome Keisler

Introduction to Calculus and Analysis - Richard Courant, Fritz John

Find the derivative of the given function in Problems 1-21.
\(f(x) = x^2\)
\(f(t) = t^2 + 3\)
\(f(x) = 1 - 2x^2\)
\(f(x) = 3x^2 + 2\)
\(f(t) = 4t\)
\(f(x) = 2 - 5x\)
\(f(t) = 4t^3\)
\(f(t)=-t^3\)
\(f(u) = 5\sqrt{u}\)
\(f(u) = \sqrt{u + 2}\)
\(g(x) = x\sqrt{x}\)
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## 3.1 How to Set Up a Problem

In applications, a calculus problem is often presented verbally, and it is up to you to set up the problem in mathematical terms. The problem can usually be described mathematically by a list of equations and inequalities. The next two sections contain several examples that illustrate the process of setting up a problem. The examples in this section are from algebra and geometry, and those in the next section are from calculus.
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## 2.8 Implicit Functions

We now turn to the topic of implicit differentiation. We say that \(y\) is an implicit function of \(x\) if we are given an equation
$$ \sigma(x, y) = \tau(x, y) $$
which determines \(y\) as a function of \(x\). An example is \(x + xy = 2y\). Implicit differentiation is a way of finding the derivative of \(y\) without actually solving for \(y\) as a function of \(x\). Assume that \(dy/dx\) exists.
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## 2.7 Higher Derivatives

DEFINITION The second derivative of a real function \(f\) is the derivative of the derivative of \(f\), and is denoted by \(f^{\prime\prime}\). The third derivative off is the derivative of the second derivative, and is denoted by \(f^{\prime\prime\prime}\), or \(f^{(3)}\). In general, the nth derivative of \(f\) is denoted by \(f^{(n)}\).
If \(y\) depends on \(x\), \(y = f(x)\), then the second differential of \(y\) is defined to be
$$ d^2y = f^{\prime\prime}(x)\ dx^2.
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## 2.6 Chain Rule

The Chain Rule is more general than the Inverse Function Rule and deals with the case where \(x\) and \(y\) are both functions of a third variable \(t\).
Suppose
$$ x = f(t), y = G(x). $$
Thus \(x\) depends on \(t\), and \(y\) depends on \(x\). But \(y\) is also a function of \(t\),
$$ y = g(t), $$
where \(g\) is defined by the rule
$$ g(t) = G(f(t)).
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## 2.5 Transcendental Functions

The transcendental functions include the trigonometric functions \(\sin x,\ \cos x,\ \tan x\), the exponential function \(e^x\), and the natural logarithm function \(\ln x\). These functions are developed in detail in Chapters 7 and 8. This section contains a brief discussion.
1 Trigonometric Functions The Greek letters \(\theta\) (theta) and \(\phi\) (phi) are often used for angles. In the calculus it is convenient to measure angles in radians instead of degrees.
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## 2.4 Inverse Functions

Two real functions \(f\) and \(g\) are called inverse functions if the two equations
$$ y = f(x), \ x = g(y) $$
have the same graphs in the \((x, y)\) plane. That is, a point \((x, y)\) is on the curve \(y = f(x)\) if, and only if, it is on the curve \(x = g(y)\). (In general, the graph of the equation \(x = g(y)\) is different from the graph of \(y = g(x)\), but is the same as the graph of \(y = f(x)\); see Figure 2.
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## 2.3 Derivatives of Rational Functions

A term of the form
$$ a_1x + a_0 $$
where \(a_1, a_0\) are real numbers, is called a linear term in \(x\); if \(a_1 \ne 0\), it is also called polynomial of degree one in \(x\). A term of the form
$$ a_2x^2 + a_1x + a_0, \ a_2 \ne 0 $$
is called a polynomial of degree two in \(x\), and, in general, a term of the form
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## 2.2 Differentials and Tangent Lines

Suppose we are given a curve \(y = f(x)\) and at a point \((a, b)\) on the curve the slope \(f^\prime(a)\) is defined. Then the tangent line to the curve at the point \((a, b)\), illustrated in Figure 2.2.1, is defined to be the straight line which passes through the point \((a, b)\) and has the same slope as the curve at \(x = a\). Thus the tangent line is given by the equation
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## 2.1 Derivatives

We are now ready to explain what is meant by the slope of a curve or the velocity of a moving point. Consider a real function \(f\) and a real number \(a\) in the domain of \(f\). When \(x\) has value \(a\), \(f(x)\) has value \(f(a)\). Now suppose the value of \(x\) is changed from \(a\) to a hyperreal number \(a + \Delta{x}\) which is infinitely close to but not equal to \(a\).
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