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. Trignometric 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. An angle \(\theta\) in radians is defined as the length of the arc of the angle on a circle of radius one (Figure \(\PageIndex{1}\)). Since a circle of radius one has circumference \(2 \pi\), \[360 \ \text{degrees} = 2\pi \ \text{radians} \nonumber\]

Figure \(\PageIndex{1}\). Angle in radians defined as length of arc on a unit circle.

Thus a right angle is \[90 \ \text{degrees} = \pi/2 \ \text{radians}. \nonumber\]

To define the sine and cosine functions, we consider a point \(P(x, y)\) on the unit circle \(x^{2} + y^{2} = 1\). Let \(\theta\) be the angle measured counterclockwise in radians from the point \((1, 0)\) to the point \(P(x, y)\) as shown in Figure \(\PageIndex{2}\). Both coordinates \(x\) and \(y\) depend on \(\theta\). The value of \(x\) is called the cosine of \(\theta\), and the value of \(y\) is the sine of \(\theta\). In symbols, \[x = \cos \theta, \quad y = \sin \theta. \nonumber\]The tangent of \(\theta\) is defined by \[\tan \theta = \sin \theta / \cos \theta. \nonumber\]

Figure \(\PageIndex{2}\): Sine and cosine functions defined using the unit circle.

Negative angles and angles greater than \(2 \pi\) radians are also allowed. The trigonometric functions can also be defined using the sides of a right triangle, but this method only works for \(\theta\) between \(0\) and \(\pi/2\). Let \(\theta\) be one of the acute angles of a right triangle as shown in Figure \(\PageIndex{3}\).

Figure \(\PageIndex{3}\): Right triangle for use in defining trigonometric functions.

Then \[\begin{align*} \sin \theta &= \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{a}{c}, \\[4pt] \cos \theta &= \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{b}{c}, \\[4pt] \tan \theta &= \frac{\text{opposite side}}{\text{adjacent side}} = \frac{a}{b}. \end{align*}\]

The two definitions, with circles and right triangles, can be seen to be equivalent using similar triangles.

Table \(\PageIndex{1}\) gives the values of \(\sin \theta\) and \(\cos \theta\) for some important values of \(\theta\).

Table \(\PageIndex{1}\). \(\sin \theta\) and \(\cos \theta\) at important values of \(\theta\).

\(\theta\) in degrees	\(0^{\circ}\)	\(30^{\circ}\)	\(45^{\circ}\)	\(60^{\circ}\)	\(90^{\circ}\)	\(180^{\circ}\)	\(270^{\circ}\)	\(360^{\circ}\)
\(\theta\) in radians	\(0\)	\(\pi/6\)	\(\pi/4\)	\(\pi/3\)	\(\pi/2\)	\(\pi\)	\(3 \pi/2\)	\(2 \pi\)
\(\sin \theta\)	\(0\)	\(1/2\)	\(\sqrt{2}/2\)	\(\sqrt{3}/2\)	\(1\)	\(0\)	\(-1\)	\(0\)
\(\cos \theta\)	\(1\)	\(\sqrt{3}/2\)	\(\sqrt{2}/2\)	\(1/2\)	\(0\)	\(1\)	\(0\)	\(1\)

A useful identity which follows from the unit circle equation \(x^{2} + y^{2} = 1\) is \[\sin^{2} \theta + \cos^{2} \theta = 1. \nonumber\]

Here \(\sin^{2} \theta\) means \((\sin \theta)^{2}\).

Figure \(\PageIndex{4}\) shows the graphs of \(\sin \theta\) and \(\cos \theta\), which look like waves that oscillate between \(1\) and \(-1\) and repeat every \(2 \pi\) radians.

The derivatives of the sine and cosine functions are: \[\begin{align*} \frac{d (\sin \theta)}{d \theta} &= \cos \theta. \\[4pt] \frac{d (\cos \theta)}{d \theta} &= - \sin \theta. \end{align*}\]

Figure \(\PageIndex{4}\): Graphs of \(\sin \theta\) and \(\cos \theta\).

In both formulas \(\theta\) is measured in radians. We can see intuitively why these are the derivatives in Figure \(\PageIndex{5}\).

In the triangle under the infinitesimal microscope, \[\begin{align*} \frac{\Delta (\sin \theta)}{\Delta \theta} &\approx \frac{\text{adjacent side}}{\text{hypotenuse}} = \cos \theta; \\[4pt] \frac{\Delta (\cos \theta)}{\Delta \theta} &\approx \frac{- \text{opposite side}}{\text{hypotenuse}} = - \sin \theta \end{align*}\]

Figure \(\PageIndex{5}\).

Notice that \(\cos \theta\) decreases, and \(\Delta (cos \theta)\) is negative in the figure, so the derivative of \(\cos \theta\) is \(- \sin \theta\) instead of just \(\sin \theta\).

Using the rules of differentiation we can find other derivatives.

Example \(\PageIndex{1}\)

Differentiate \(y = \sin^{2} \theta\).

Solution

Let \(u = \sin \theta\), \(y = u^{2}\). Then \[\frac{dy}{d \theta} = 2u \frac{du}{d \theta} = 2 \sin \theta \cos \theta. \nonumber\]

Example \(\PageIndex{2}\)

Differentiate \(y = \sin \theta (1 - \cos \theta)\).

Solution

Let \(u = \sin \theta\), \(v = 1 - \cos \theta\). Then \(y = u \cdot v\), and \[\begin{align*} \frac{dy}{d \theta} = u \frac{dv}{d \theta} + v \frac{du}{d \theta} &= \sin \theta (-(- \sin \theta)) + (1 - \cos \theta) \cos \theta \\ &= \sin^{2} \theta + \cos \theta - \cos^{2} \theta. \end{align*}\]

The other trigonometric functions (the secant, cosecant, and cotangent functions) and the inverse trigonometric functions are discussed in Chapter 7.

2. Exponential Functions

Given a positive real number \(b\) and a rational number \(m/n\), the rational power \(b^{m/n}\) is defined as \[b^{m/n} = \sqrt[n]{b^{m}}, \nonumber\]the positive \(n\)th root of \(b^{m}\). The negative power \(b^{-m/n}\) is \[b^{-m/n} = \frac{1}{b^{m/n}}. \nonumber\]

As an example consider \(b = 10\). Several values of \(10^{m/n}\) are shown in Table \(\PageIndex{2}\).

Table \(\PageIndex{2}\).

\(10^{-3}\)	\(10^{-3/2}\)	\(10^{-1}\)	\(10^{-2/3}\)	\(10^{-1/3}\)	\(10^{0}\)	\(10^{1/3}\)	\(10^{2/3}\)	\(10^{1}\)	\(10^{3/2}\)	\(10^{3}\)
\(\dfrac{1}{1000}\)	\(\dfrac{1}{10 \sqrt{10}}\)	\(\dfrac{1}{10}\)	\(\dfrac{1}{\sqrt[3]{100}}\)	\(\dfrac{1}{\sqrt[3]{10}}\)	\(1\)	\(\sqrt[3]{10}\)	\(\sqrt[3]{100}\)	\(10\)	\(10 \sqrt{10}\)	\(1000\)

If we plot all the rational powers \(10^{m/n}\), we get a dotted line, with one value for each rational number \(m/n\), as in Figure \(\PageIndex{6}\).

Figure \(\PageIndex{6}\): Plotted values of several rational \(x\) values of \(10^{x}\).

By connecting the dots with a smooth curve, we obtain a function \(y = 10^{x}\), where \(x\) varies over all real numbers instead of just the rationals. \(10^{x}\) is called the
exponential function with base 10. It is positive for all \(x\) and follows the rules \[10^{a + b} = 10^{a} \cdot 10^{b}, \quad 10^{a \cdot b} = \left(10^{a}\right)^{b}. \nonumber\]

The derivative of \(10^{x}\) is a constant times \(10^{x}\), approximately \[\frac{d \left(10^{x}\right)}{dx} \sim (2.303) 10^{x}. \nonumber\]

To see this let \(\Delta x\) be a nonzero infinitesimal. Then \[\frac{d \left(10^{x}\right)}{dx} = st \left[\frac{10^{x + \Delta x} - 10^{x}}{\Delta x}\right] = st \left[\frac{\left(10^{x} - 1\right) 10^{x}}{\Delta x}\right] = st \left[\frac{10^{\Delta x} - 1}{\Delta x}\right] 10^{x}. \nonumber\]

The number \(st \left[ \left(10^{\Delta x} - 1\right)/\Delta x\right]\) is a constant which does not depend on \(x\) and can be shown to be approximately \(2.303\).

If we start with a given positive real number \(b\) instead of \(10\), we obtain the exponential function with base \(b\), \(y = b^{x}\). The derivative of \(b^{x}\) is equal to the constant \(st \left[ \left(b^{\Delta x} - 1\right)/\Delta x\right]\) times \(b^{x}\). This constant depends on \(b\). The derivative is computed as follows: \[\frac{d \left(b^{x}\right)}{dx} = st \left[\frac{b^{x + \Delta x} - b^{x}}{\Delta x}\right] = st \left[\frac{\left(b^{\Delta x} - 1\right) b^{x}}{\Delta x}\right] = st \left[\frac{b^{\Delta x} - 1}{\Delta x}\right] b^{x}. \nonumber\]

The most useful base for the calculus is the number \(e\). \(e\) is defined as the real number such that the derivative of \(e^{x}\) is \(e^{x}\) itself, \[\frac{d \left(e^{x}\right)}{dx} = e^{x}. \nonumber\]

In other words, \(e\) is the real number such that the constant \[st \left[\frac{e^{\Delta x} - 1}{\Delta x}\right] = 1. \nonumber\](where \(\Delta x\) is a nonzero infinitesimal). It will be shown in Section 8.3 that there is such a number \(e\) and that \(e\) has the approximate value \[e \sim 2.71828. \nonumber\]

The function \(y = e^{x}\) is called the exponential function. \(e^{x}\) is always positive and follows the rules \[e^{a + b} = e^{a} \cdot e^{b}, \quad e^{a \cdot b} = \left(e^{a}\right)^{b}, \quad e^{0} = 1. \nonumber\]

Figure \(\PageIndex{7}\) shows the graph of \(y = e^{x}\).

Figure \(\PageIndex{7}\): Graph of \(y = e^{x}\).

Example \(\PageIndex{3}\)

Find the derivative of \(y = x^{2} e^{x}\).

Solution

By the Product Rule, \[\frac{dy}{dx} = x^{2} \frac{d \left(e^{x}\right)}{dx} + e^{x} \frac{d \left(x^{2}\right)}{dx} = x^{2} e^{x} + 2x e^{x}. \nonumber\]

3. The Natural Logarithm

The inverse of the exponential function \(x = e^{y}\) is the natural logarithm function, written \[y = \ln x. \nonumber\]

Verbally, \(\ln x\) is the number y such that \(e^{y} = x\). Since \(y = \ln x\) is the inverse function of \(x = e^{y}\), we have \[e^{\ln a} = a, \quad \ln \left(e^{a}\right) = a. \nonumber\]

The simplest values of \(y = \ln x\) are \[\ln (1/e) = -1, \quad \ln (1) = 0, \quad \ln e = 1. \nonumber\]

Figure \(\PageIndex{8}\) shows the graph of \(y = \ln x\). It is defined only for \(x > 0\).

Figure \(\PageIndex{8}\): Graph of \(y = \ln x\).

The most important rules for logarithms are \[\begin{align*} \ln(ab) &= \ln a + \ln b, \\ \ln (ab) &= b \cdot \ln a. \end{align*}\]

The natural logarithm function is important in calculus because its derivative is simply \(1/x\), \[\frac{d (\ln x)}{dx} = \frac{1}{x}, \quad (x > 0). \nonumber\]

This can be derived from the Inverse Function Rule.

If \(y = \ln x\), then \[\begin{align*} x &= e^{y}, \\ \frac{dx}{dy} &= e^{y}, \\ \frac{dy}{dx} &= \frac{1}{dx/dy} = \frac{1}{e^{y}} = \frac{1}{x}. \end{align*}\]

The natural logarithm is also called the logarithm to the base \(e\) and is sometimes written \(\log_{e} x\). Logarithms to other bases are discussed in Chapter 8.

Example \(\PageIndex{4}\)

Differentiate \(y = \dfrac{1}{\ln x}\).

Solution

\[\frac{dy}{dx} = \frac{-1}{(\ln x)^{2}} \frac{d (\ln x)}{dx} = \frac{-1}{x (\ln x)^{2}}. \nonumber\]

4. Summary

Here is a list of the new derivatives given in this section. \[\begin{align*} \frac{d (\sin x)}{dx} &= \cos x, \\[4pt] \frac{d (\cos x)}{dx} &= - \sin x, \\[4pt] \frac{d e^{x}}{dx} &= e^{x}, \\[4pt] \frac{d (\ln x)}{dx} &= \frac{1}{x} \quad (x > 0). \end{align*}\]

Tables of values for \(\sin x\), \(\cos x\), \(e^{x}\), and \(\ln x\) can be found at the end of the book.

Problems for Section 2.5

In Problems 1-20, find the derivative.

1.	\(y = \cos^{2} \theta\)	2.	\(s = \tan^{2} t\)
3.	\(y = 2 \sin x + 3 \cos x\)	4.	\(y = \sin x \cdot \cos x\)
5.	\(w = \frac{1}{\cos z}\)	6.	\(w = \dfrac{1}{\sin z\)
7.	\(y = \sin^{n} \theta\)	8.	\(y = \tan^{n} \theta\)
9.	\(s = t \sin t\)	10.	\(s = \dfrac{\cos t}{t - 1}\)
11.	\(y = xe^{x}\)	12.	\(y = 1/\left(1 + e^{x}\right)\)
13.	\(y = (\ln x)^{2}\)	14.	\(y = x \ln x\)
15.	\(y = e^{x} \cdot \ln x\)	16.	\(y = e^{x} \cdot \sin x\)
17.	\(u = \sqrt{v} \left(1 - e^{n}\right)\)	18.	\(u = \left(1 + e^{v}\right) \left(1 - e^{v}\right)\)
19.	\(y = x^{n} \ln x\)	20.	\(y = (\ln x)^{n}\)

In Problems 21-24, find the equation of the tangent line at the given point.

21.	\(y = \sin x \text{ at } (\pi/6, 1/2)\)	22.	\(s = \cos x \text{ at } (\pi/4, \sqrt{2}/2)\)
23.	\(y = x - \ln x \text{ at } (e, e-1)\)	24.	\(y = e^{-x} \text{ at } (0, 1)\)