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 1: The Derivative
The Definition of Average Rate of Change
The average rate of change of a function $f(x)$ over an interval $a\le x\le b$ is defined to be
$$
\frac {f(b)-f(a)}{b-a}
$$
Geometrically
In geometrical, the average rate of change is the slope of the secant line through the points $(a,f(a))$ and $(b,f(b))$.
The Definition of the Derivative
The derivative of a function $f(x)$ at a point $x=a$ is defined as
$$
f^\prime(a)=\lim_{b\to a}\frac{f(b)-f(a)}{b-a}
$$
Geometrically
In geometrical, the derivative $f^\prime(a)$ is the slope of the tangent line to the function $f$ through the point $(a, f(a))$.
Property of Derivatives
The derivative of a function is itself a function, and satsifies the following linearity properties:
Derivatives of Constant Multiples
If $g(x)=k\cdot f(x)$ for some constant $k$, then $$ g^\prime(x)=k\cdot f^\prime(x) $$ at all points where $f$ is differentiable.
Derivatives of Sums
If $h(x)=f(x)+g(x)$, then $$ h^\prime(x)=g^\prime(x)+f^\prime(x) $$ at all points where $f$ and $g$ are differentiable.
Derivatives of Differences
If $h(x)=f(x)-g(x)$, then $$ h^\prime(x)=f^\prime(x)-g^\prime(x) $$ at all points where $f$ and $g$ are differentiable.
The Power Rule
If $n$ is any fixed real numberm, and $f(x)=x^n$, then $f^\prime(x)=n\cdot x^{n-1}$.
Properties of Leibniz Notation
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Unit: Suppose $P$ has units of pressure and $t$ has units of time. Then the derivative $$ \frac{dP}{dt} $$ has units of pressure per unit time.
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Evaluating at Points: To denote the derivative of a function at a particular point, for example at $x=3$, we write $$ \left.\frac{df}{dx}\right|_{x=3}, $$ where the ertical bar indicates that the derivative is evaluated at $x=3$.
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Derivatives Acting on Functions:
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One may write $$ \frac{d}{dx}\left(x^2\right) $$ for the derivative of $x^2$.
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Alternatively, we may write $$ \frac{d}{dy}\left(y^3+2y^2\right). $$ for a more complex function $y^3+2y^2$.
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Second Derivative
The second derivative of a function $f(x)$ is defined as the derivative of $f^\prime(x)$, and is denoted by either $$ f^{\prime\prime}(x) $$ or $$ \frac{d^2 f}{dx^2} $$ Notice that in the Leibniz Notation for the second derivative, the square in the demoniator should be marked on the $d$, while in the numerator, the squre should be placed after $x$.
Second Derivative and Concavity Summary
On intervals where $f^{\prime\prime}>0$, the function $f$ is concave up.
On intervals where $^{\prime\prime}<0$, the function $f$ is concave down.
Derivative of Sine and Cosine
$$ \begin{aligned} \frac d {dx} sin(x)\quad&=\quad cos(x)\\ \frac d {dx} cos(x)\quad&=\quad-sin(x)\\ \frac {d^2} {dx^2} sin(x)\quad&=\quad -sin(x)\\ \frac {d^2} {dx^2} cos(x)\quad&=\quad -cos(x) \end{aligned} $$