Preface
This is the note of the MIT Course 18.01.1 Calculus. For more details, refer to MIT 18.x Catalog, and it can also be found at MIT Open Learning Library.
Unit 2: Differentiation
Linear Approximation
The linear approximation for a function $f$ near a point $x=a$ is given by the following equivalent formulas: $$ \begin{aligned} \Delta f &\approx \left. \frac{df}{dx}\right |_{x=a} \cdot \Delta x\quad &for\ \Delta x\ near\ 0\\ f(x)&\approx f^\prime(a)\cdot(x-a)+f(a)\quad &for\ x\ near\ a \end{aligned} $$
The Product Rule
If $h(x)=f(x)\cdot g(x)$, then $$ h^\prime(x)=f(x)\cdot g^\prime(x)+f^\prime(x)\cdot g(x) $$ at all points where the derivatives $f^\prime(x)$ and $g^\prime(x)$ are defined.
The Quotient Rule
If $h(x)=\frac{f(x)}{g(x)}$ for all $x$, then $$ h^\prime(x)=\frac{f^\prime(x) \cdot g(x)-f(x)\cdot g^\prime(x)}{g(x)^2} $$ at all points where $f$ and $g$ are differentiable and $g(x)\ne 0$.
The Chain Rule
If $h(x)=f(g(x))$, then $$ h^\prime(x)=f^\prime(g(x))\cdot g^\prime(x) $$ at all points where the derivatives $f^\prime(g(x))$ and $g^\prime(x)$ are defined.
Alternatively, if $y=f(u)$, and $u=g(x)$, then $$ \left.\frac{dy}{dx}\right\vert_{x=a}= \left. \frac{dy}{du}\right\vert_{u=g(a)} \cdot \left. \frac{du}{dx}\right\vert_{x=a} $$ at any point $x=a$ where the derivatives $\left.\frac{dy}{du}\right\vert_{u=g(a)}$ and $\left.\frac{du}{dx}\right\vert_{x=a}$ are defined.
Implicit Differentiation
To implicitly differentiate a function $f(x)\cdot g(x)=1$ with respect to $x$: $$ \begin{aligned} &\frac d {dx}(f(x)\cdot g(y)\quad &=& \quad 1)\\ &\frac d {dx}(f(x)\cdot g(y))\quad &=& \quad \frac d {dx}1\\ &\frac d{dx}(f(x)\cdot g(y))\quad &=& \quad 0\\ &f^\prime(x)\cdot g(x)+f(x)\cdot\frac{d}{dx}g(y)\quad &=& \quad 0\\ &f^\prime(x)\cdot g(x)+f(x)\cdot g^\prime(y)\cdot\frac{dy}{dx}\quad &=& \quad 0 \end{aligned} $$
Definition of Inverser Function
If functions $f$ and $g$ satisfy $g(f(x))=x$ and $f(g(y))=y$, then we say $g$ is the inverse of $f$, and denote it by $f^{-1}$. Similarly, $f=g^{-1}$.
If a function $f$ has an inverse function $f^{-1}$, then $f^{-1}(b)=a$ if and only if $f(a)=b$.
Definition of One-to-One
A function $f$ is one-to-one if $f(a)\ne f(b)$ whenever $a\ne b$. It is one-to-one if and only if its graphy satisfies the horizontal line test(noi horizontal line intersects its graph at more than one place).
Domain and Range
Recall that the domain of a function $f$ is the set of allowable input values. For instance, the domain of the function $f(x)=\frac 1 x$ si the set of all non-zero real numbers.
The range of $f$ is the set of all possible output values. For instance, the range of the function $g(x)=x^2$ is the set of all real numbers that are non-negative.
Derivatives of Inverse Functions
If $g$ is a inverse of a function $f$, then $$ g^\prime = \frac 1 {f^\prime (g(x))} $$ at all $x$ where $f^\prime(g(x))$ exists and is non-zero.
Derivatives of the Inverse Trig Functions
We now have more basic functions that we can differentiate. $$ \begin{aligned} \frac d {dx}\arcsin x\quad&=&\quad \frac 1 {\sqrt{1-x^2}}\\ \frac d {dx}\arccos x\quad&=&\quad -\frac 1 {\sqrt{1-x^2}}\\ \frac d {dx}\arctan x\quad&=&\quad \frac 1 {1+x^2} \end{aligned} $$
The Derivative of an exponential function
The derivative of the exponential function is $$ \frac d {dx}a^x=M(a)a^x $$ where the mystery number $M(a)$ is the slope of the tangent line at zero: $$ M(a)=\left.\frac d {dx}a^x\right\vert_{x=0}=\lim_{\Delta x\to 0}\frac{a^{\Delta x}-1}{\Delta x} $$
Definition of $e$
The base $e$ is the unique real number so that $$ \frac d{dx}e^x=e^x $$
Differentiating Exponential Function with other Bases
We can finally identify out mystery number, and differentiate exponential functions with any base.
For any postivie constant $a$, $$ \frac d{dx}a^x=a^x\ln a $$
Properties of $x$
$\log_a(x)$ is the inverse function of $ a^x$.
The natural log, denoted $\ln(x)$, is the inverse function of $e^x$.
We have
- $\ln e^x=x$
- $e^{\ln x}=x$
- $\ln(ab)=\ln(a)+\ln(b)$
- $\ln(a^b)=b\cdot\ln(a)$
The Derivative of the Natural Logarithm
Now we can differentiate more basic function: $$ \frac d {dx}\ln x=\frac 1 x $$