Transcendental Functions
Exponential, logarithm, and trigonometric functions, and their inverses — defined carefully and differentiated. The clean way to build them is to define $\ln$ as an integral and recover everything else from there.
- Define $\ln$ and $\exp$ rigorously and prove their basic laws.
- Differentiate exponential, logarithmic, and trig functions.
- Differentiate inverse trig functions.
- Use logarithmic differentiation.
Polynomials cannot capture growth that compounds or motion that repeats. Exponentials, logarithms, and trig functions can — they are called "transcendental" because they escape algebra entirely.
$e^x$ models compound interest, radioactive decay, and populations; logarithms give us pH, decibels, the Richter scale, and the "bit" of information; trig functions underlie all signal processing — audio, JPEG images, MRI, and GPS all run on Fourier analysis built from sines and cosines.
Napier invented logarithms in 1614 to shorten astronomers' brutal hand calculations. Euler named $e$ and tied it all together. Later $e$ and $\pi$ were proved transcendental (1873, 1882), which finally settled the 2,000-year-old puzzle of whether you can "square the circle." You cannot.
How the functions are built
The clean MAT137 approach is structural: define $\ln$ by an integral, prove its laws, define $\exp$ as its inverse, then define $a^x$ using $e^{x\ln a}$. This order matters because it keeps irrational exponents and logarithm laws from becoming circular assumptions.
Domain discipline
Most mistakes in this unit are not algebra mistakes; they are domain mistakes. Before differentiating, ask where the expression is defined and where the derivative formula is valid.
- $\ln x$ requires $x>0$, while $\ln|x|$ is defined for $x\ne0$.
- $a^x=e^{x\ln a}$ requires $a>0$ for real-valued functions of all real $x$.
- $\arcsin x$ is defined on $[-1,1]$, but its derivative formula only holds on $(-1,1)$.
- Logarithmic differentiation requires the quantity inside $\ln$ to be positive, or else an absolute-value version must be justified.
The natural logarithm as an integral
For $x>0$, $\displaystyle\ln x=\int_1^x \frac{1}{t}\,dt.$ By the Fundamental Theorem of Calculus, $\dfrac{d}{dx}\ln x=\dfrac1x$.
For $a,b>0$: $\ln(ab)=\ln a+\ln b$ and $\ln(a/b)=\ln a-\ln b$. For rational $r$, $\ln(a^r)=r\ln a$; after real powers are defined by $a^r=e^{r\ln a}$, the same identity holds for every real $r$.
The exponential function
$\exp=\ln^{-1}$, and $e=\exp(1)$. Then $\dfrac{d}{dx}e^x=e^x$, and for $a>0$, $a^x=e^{x\ln a}$ with $\dfrac{d}{dx}a^x=a^x\ln a$.
The derivative rule follows from the inverse function theorem (Unit 3 / next units): if $y=e^x$ then $\ln y=x$, so $\dfrac{y'}{y}=1$, giving $y'=y=e^x$.
Trigonometric functions
$\displaystyle\lim_{x\to0}\frac{\sin x}{x}=1$ — the fact behind every trig derivative. From it, $\dfrac{d}{dx}\sin x=\cos x$ and $\dfrac{d}{dx}\cos x=-\sin x$.
$\dfrac{d}{dx}\tan x=\dfrac{d}{dx}\dfrac{\sin x}{\cos x}= \dfrac{\cos^2x+\sin^2x}{\cos^2x}=\sec^2x.$
Inverse trigonometric functions
$\dfrac{d}{dx}\arcsin x=\dfrac{1}{\sqrt{1-x^2}}$ for $|x|<1$, $\dfrac{d}{dx}\arctan x=\dfrac{1}{1+x^2}$ for every real $x$.
Logarithmic differentiation
For products, quotients, and variable exponents, take $\ln$ first.
Differentiate $y=x^x$ ($x>0$).
• $\dfrac{d}{dx}a^x\ne x a^{x-1}$. The power rule is for $x^{n}$, not $a^x$.
• Mixing up $(\ln x)'=1/x$ with $(\log_{10}x)'=1/(x\ln 10)$.
• Forgetting domains: $\arcsin x$ needs $|x|\le1$, and its derivative formula
requires $|x|<1$.
Practice
Problems adapted from the MAT137 question bank by Alfonso Gracia-Saz, used under CC BY-SA 4.0. Write full solutions before opening the answer.
Differentiate $f(x)=\ln(\sec x+\tan x)$ where it is defined.
Answer
Show $\dfrac{d}{dx}\arcsin x=\dfrac{1}{\sqrt{1-x^2}}$ for $|x|<1$.
Hint
Obtain and prove the formula for $\dfrac{d}{dx}\arctan x$.
Solution
Find the derivative of $F(x)=\ln|x|$.
Answer
Differentiate $f(x)=e^{\sin x+\cos x}\ln x$ for $x>0$.
Answer
Use logarithmic differentiation on $y=\dfrac{(x+1)^2\sqrt{x-3}}{x^4}$.
Hint
Calculate the derivative of $g(x)=x^{\tan x}$ for $x>0$.
Solution
Assume $x,y>0$. Find $y'$ if $x^y=y^x$.
Solution
Expanded topic checklist
Course coverage. This unit is part of the complete MAT137 topic map. It should cover the following items before you move on:
- Logarithm and exponential. Know how $\ln x$ and $e^x$ are defined, why they are inverses, and how their derivative rules follow.
- Trigonometric functions. Use standard trig identities and derivative rules without losing domain restrictions.
- Inverse trig. Differentiate inverse trigonometric functions through inverse-function reasoning.
- Logarithmic differentiation. Use logarithms to differentiate powers, products, quotients, and variable-base expressions.
Use the Alfonso Gracia-Saz MAT137 materials as the primary local spine, with the current UofT single-variable calculus topic split as the completeness check. The added items are original study prompts, not copied textbook exercises.
The spine is: define one function rigorously, prove it is monotone and invertible, then use the inverse derivative rule to obtain the companion derivative formula.
Deep practice extensions
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
Starting from $\ln x=\int_1^x 1/t\,dt$, explain why $(\ln x)\prime=1/x$ for $x>0$.
Guide
Use inverse differentiation to show $(e^x)\prime=e^x$ if $e^x$ is the inverse of $\ln x$.
Guide
Derive $(rcsin x)\prime=1/\sqrt{1-x^2}$ for $-1<x<1$.
Guide
Differentiate $y=x^x$ for $x>0$.