# WHY: Why should we learn differentiation?

How to find the speed? How to find the acceleration? - Find the rate: At (when), the rate at which (how) is increasing/decreasing by (what) (unit).

How to sketch a function? - Find the tendency of a function, knowning monotonicity and concavity. (& Optimization)

# WHAT: What is differentiation?

$$ \lim_{h \rightarrow 0}\frac{f(x + h) - f(x)}{h} \textbf{ (1)} = \lim_{\Delta{x} \rightarrow 0}\frac{\Delta{f(x)}}{\Delta{x}} \textbf{ (2)} = \frac{\mathrm{d}f(x)}{\mathrm{d}x} \textbf{ (3)} \Rightarrow f’(x) \textbf{ (4)} $$

Notice. When transform from #2 to #3, we make two simplifications: Change $\Delta$ to $\mathrm{d}$; Leave out $\lim_{\Delta{x} \rightarrow 0}$. So that is a reason why $\frac{\mathrm{d}f(x)}{\mathrm{d}x} \neq \mathrm{d}f(x) \div \mathrm{d}x$, because $f(x)$ and $x$ are not in the same manner, or $\frac{\mathrm{d}f(x)}{\mathrm{d}x}$ is respected to $x$. Thus, it is preferred to write in $\frac{d}{\mathrm{d}x}f(x)$, or $f’(x)$ for reminding which is respected.

Difference between differentiation and derivative?

• $\frac{\mathrm{d}f(x)}{\mathrm{d}x}$ differentiation is a method to crop source function to small range for knowing the tendency
• $f’(x)$ derivative is that kinds of tendency.

Or we can say, differentiation is a method to find the derivative.

EX-CAL-DIF-01: Use the derivative way to present $g(x)$: $g(x) = \lim_{h \rightarrow 0}\Large{\frac{f(x + h) - \frac{3}{4}f(x + 2h) + \frac{1}{8}f(x + 4h) - \frac{3}{8}f(x)}{h^3}}$

# WHY: Why a function could be differentiated? - Differentiability

• Limit & Continuity: $\lim_{x \rightarrow a}{f(x)} = f(a)$ ($\lim_{x \rightarrow a}{f(x)} = \lim_{x \rightarrow a^+}{f(x)} = \lim_{x \rightarrow a^-}{f(x)}$)
  • IVT (Intermediate Value Theorem): If $f(x)$ is continuous on $[a, b]$, then $\exists c \in [a,b], f(c) \in [f(a), f(b)]$
    • MVT (Mean Value Theorem): If $f(x)$ is differentiable on $[a, b]$, then $\exists c, f’(c) = \frac{f(b) - f(a)}{b - a}$
    • EVT (Extreme Value Theorem): If $f(x)$ is continuous on closed interval $[a, b]$, then $\exists c_1, c_2, f_{max}(x) = f(c_1), f_{min}(x) = f(c_2)$
  • Discontinuous points (Discontinuity)
    • Removable ~: $\lim_{x \rightarrow a}f(x) \neq f(a)$
    • Jump ~: $\lim_{x \rightarrow a+}f(x) \neq \lim_{x \rightarrow a-}f(x)$
    • Infinite (Essential) ~: $\lim_{x \rightarrow a^+}f(x)$ or $\lim_{x \rightarrow a^-}f(x)$ equals $\pm \infty$
• Smooth: $\lim_{c \rightarrow 0^+}{\frac{f(a + c) - f(c)}{c}} = \lim_{c \rightarrow 0^-}{\frac{f(a + c) - f(c)}{c}}$
  • e.i. L’Hôpital’s rule

Does EX-CAL-DIF-01 have an easier way to solve?

EX-CAL-DIF-02-01: $\lim_{x \rightarrow 0^+}{x\ln{x}}$, $\lim_{x \rightarrow 0^+}{x^x}$

EX-CAL-DIF-02-02: $\lim_{x \rightarrow \pm\infty}(1 + \frac{1}{x})^x$ (Important one, that’s the definition of $e$)

EX-CAL-DIF-02-03: $\lim_{x \rightarrow 0}\frac{\sin{x} - x}{x(\cos{x} - 1)}$

# WHAT: What is the derivative for basic function

$$ \frac{d}{\mathrm{d}x}x^n = n \cdot x^{n - 1}, \frac{d}{\mathrm{d}x}Constant = 0, \frac{d}{\mathrm{d}x}\ln{x} = \frac{1}{x}, \frac{d}{\mathrm{d}x}e^x = e^x $$ $$ \frac{d}{\mathrm{d}x}\sin{x} = \cos{x}, \frac{d}{\mathrm{d}x}\cos{x} = -\sin{x}, \frac{d}{\mathrm{d}x}\tan{x} = \frac{1}{\cos^2{x}} = \sec^2{x} $$ $$ \frac{d}{\mathrm{d}x}\sin^{-1}{x} = \frac{d}{\mathrm{d}x}\arcsin{x} = \frac{1}{\sqrt{1 - x^2}}, \frac{d}{\mathrm{d}x}\cos^{-1}{x} = -\frac{1}{\sqrt{1 - x^2}}, \frac{d}{\mathrm{d}x}\tan^{-1}{x} = \frac{1}{1 + x^2} $$

EX-CAL-DIF-03: Try to use definition to derive the first two lines.

# HOW: How to differentiate? - Basic rules

$$ \frac{d}{\mathrm{d}x}(f(x) + g(x)) = \frac{d}{\mathrm{d}x}f(x) + \frac{d}{\mathrm{d}x}g(x) \Leftrightarrow (f + g)’ = f’ + g' $$ $$ \frac{d}{\mathrm{d}x}(f(x) \cdot g(x)) = \frac{d}{\mathrm{d}x}f(x) \cdot g(x) + f(x) \cdot \frac{d}{\mathrm{d}x}g(x) \Leftrightarrow (f \cdot g)’ = f’ \cdot g + f \cdot g' $$ $$ \text{Chain rule: } \frac{d}{\mathrm{d}x}f(g(x)) = f’(g(x)) \cdot g’(x) = \frac{d}{dg(x)}f(g(x)) \cdot \frac{d}{\mathrm{d}x}g(x) $$ $$ \text{Inverse function: }y(x) \Leftrightarrow x(y) \Rightarrow \frac{d}{\mathrm{d}x}y(x_1) \cdot \frac{d}{\mathrm{d}x}x(y_1) = 1 $$

EX-CAL-DIF-04-01: Try to use the basic rule to derive the third line in WHAT: What is the derivative for basic function

EX-CAL-DIF-04-02: Differentiate: $a^x$; $\log_ax$; $x^x$

EX-CAL-DIF-04-03: Known $x^2 + y^2 = 25$, derive $\frac{d}{\mathrm{d}x}y$

# HOW: How to apply differentiation in solving problem?

# Finding Rates

EX-CAL-DIF-05-01: A particle moves along the x-axis so that at time $t \geq 0$ the position of the particle is given by $x(t) = 0.5t^4 - 1.5t^3 - 2t^2 + 6t - 1$. What is the velocity of the particle at the first instance the particle is at the origin?

EX-CAL-DIF-05-02: The sides and diagonal of the rectangle are strictly increasing with time. At the instant when two sides of rectangle $x = 4$ and $y = 3$, $\frac{\mathrm{d}}{\mathrm{d}t}x = \frac{\mathrm{d}}{\mathrm{d}t}z$ and the rate of the diagonal $z$ has $\frac{\mathrm{d}}{\mathrm{d}t}z = \frac{\mathrm{d}}{\mathrm{d}t}y$. Derive the value of $k$.

# Curve Sketching & Optimization

• Monotonicity with $\mathrm{d}y/\mathrm{d}x$
  • Monotonic increasing - $\mathrm{d}y/\mathrm{d}x > 0$
  • Monotonic decreasing: $\mathrm{d}y/\mathrm{d}x < 0$
  • Critical point: $f’(x) = 0$ or $\nexists f’(x)$, Stationary point: $f’(x) = 0$, Relative/Local Extrema: $f’(x) = 0$ and $f’’(x) \neq 0$
• Concavity with $d^2y/\mathrm{d}x^2$
  • Concave upward: $d^2y/\mathrm{d}x^2 > 0$
  • Concave downward: $d^2y/\mathrm{d}x^2 < 0$
  • Inflected point: $f’’(x) = 0$ and $f^{(3)}(x) \neq 0$