Week 5 - Techniques of Differentiation #

The Product Rule #

• If f and g are both differentiable, then $$\frac{d}{dx}[f(x)g(x)]=f(x)\frac{d}{dx}g(x) + \frac{d}{dx}f(x)g(x)$$
• Two ways to prove

  • the area of rectangle

    • 

    • We start by assuming that u = f(x) and v = g(x).

    • Then we can interpret the product uv as an area of a rectangle

    • If x changes by an amount $\Delta x$, then the corresponding changes in u and v are

      • $\Delta u = f(x + \Delta x) - f(x)$, $\Delta v = g(x + \Delta x) - g(x)$

    • The change in the area of the rectangle is:

      • $\Delta (uv) = (u + \Delta u)(v + \Delta v) - uv = u\Delta v + v\Delta u + \Delta u\Delta v$ = the sum of the three shaded areas

    • If we divide by $\Delta x$, we get $$\frac{\Delta (uv)}{\Delta x} = \frac{u\Delta v}{\Delta x} + \frac{v\Delta u}{\Delta x} + \frac{\Delta u\Delta v}{\Delta x}$$

    • If we now let $\Delta x \to 0$, we get the derivative of uv(f(x)g(x)):

      $$\begin{aligned} \frac{d}{dx}(uv) &= \lim_{\Delta x \to 0}\frac{\Delta (uv)}{\Delta x} \\ &= \lim_{\Delta x \to 0}(u\frac{\Delta v}{\Delta x} + v\frac{\Delta u}{\Delta x} + \Delta u\frac{\Delta v}{\Delta x}) \\ &= u\lim_{\Delta x \to 0}\frac{\Delta v}{\Delta x} + v\lim_{\Delta x \to 0}\frac{\Delta u}{\Delta x} + (\lim_{\Delta x \to 0}\Delta u)(\lim_{\Delta x \to 0}(\frac{\Delta v}{\Delta x}) \\ &= u\frac{d}{dx}v + v\frac{d}{dx}u + 0 \cdot u\frac{d}{dx}v \end{aligned}$$

    • $\frac{d}{dx}(uv) = u\frac{d}{dx}v + v\frac{d}{dx}u$

  • use limit theorem

    $$\begin{aligned} \frac{d}{dx}(f(x) \cdot g(x)) &= \lim_{h \to 0}\frac{f(x+h) \cdot g(x+h)-f(x) \cdot g(x)}{h} \\ &= \lim_{h \to 0}\frac{f(x+h) \cdot g(x+h)-f(x+h) \cdot g(x)+f(x+h)g(x)-f(x) \cdot g(x)}{h} \\ &= \lim_{h \to 0}\frac{f(x+h) \cdot g(x+h)-f(x+h) \cdot g(x)}{h} + \lim_{h \to 0}\frac{f(x+h) \cdot g(x)-f(x) \cdot g(x)}{h} \\ &= \lim_{h \to 0}\frac{g(x+h)-g(x)}{h} \cdot \lim_{h \to 0}f(x+h) + \lim_{h \to 0}\frac{f(x+h)-f(x)}{h} \cdot \lim_{h \to 0}g(x) \\ &= \frac{d}{dx}g(x)\lim_{h \to 0}f(x+h) + \frac{d}{dx}f(x)\lim_{h \to 0}g(x) \\ &= f(x)\frac{d}{dx}g(x) + g(x)\frac{d}{dx}f(x) \\ \end{aligned}$$

  • In prime notion: $(f \cdot g)’=f \cdot g’+g \cdot f'$

The Quotient Rule #

• If f and g are differentiable, then

  • $$\frac{d}{dx}[\frac{f(x)}{g(x)}]=\frac{\frac{d}{dx}f(x) \cdot g(x) - f(x) \cdot \frac{d}{dx}g(x)}{g(x)^2}$$

  • In prime notion: $(\frac{f}{g})’=\frac{g \cdot f’-f \cdot g’}{g^2}$

Higher Derivatives #

• If f is a differentiable function, then its derivative f' is also a function, so f' may have a derivative of its own, denoted by (f')' = f''.
• This new function $f’’(x)$ is called the second derivative of f because it is the derivative of the derivative of f. Using Leibniz notation, we write the second derivative of y = f(x) as
  • 
• For example:
  • $f(x) = x^3 - x$
  • $f’(x)=3x^2-1$, $f’’(x)=6x$
  • 
  • We can interpret f''(x) as the slope of the curve y=f'(x) at the point (x, f(x)). In other words, it is the rate of change of the slope of the original curve y=f(x).
• Take another sample:
  • If s=s(t) is the position function of an object that moves in a straight line, we know that its first derivative represents the velocity v(t) of the object as a function of time:
    • $v(t)=s’(t)=\frac{ds}{dt}$
  • Thus the acceleration(the instantaneous rate of change of velocity with respect to time) function is the derivative of the velocity function and is therefore the second derivative of the position function:
    • $a(t)=v’(t)=s’’(t)$
    • or in Leibniz notation: $a=\frac{dv}{dt}=\frac{d^2s}{dt^2}$

Concavit #

• If f''(x) > 0 for all x in I, then the graph of f is concave upward on I.
• If f''(x) < 0 for all x in I, then the graph of f is concave downward on I.
• inflection point
  • A point P on a curve y = f(x) is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P.

Extreme Value #

Definitions #

• local maximum
  • f(c) is a local maximum value for f if there is some ε > 0, so that whenever x is in (c - ε, c + ε), f(c) >= f(ε).
• local minimum
  • f(c) is a local minimum value for f if there is some ε > 0, so that whenever x is in (c - ε, c + ε), f(c) <= f(ε).
• local extremum
  • a local maximum or local minimum is called a local extremum.
• global maximum
  • f(c) is a global maximum value for f if whenever x is in the domain of f, f(c) >= f(ε).
• global minimum
  • f(c) is a global minimum value for f if whenever x is in the domain of f, f(c) <= f(ε).

Fermat’s Theorem (Find Extreme Value) #

• Suppose f is a function, defined on the interval $(a, b) \ni c$.
  • If f(c) is an extreme value of f, and f is differentiable at c, then f'(c) = 0;
  • If f is differentiable at c, and f'(c) != 0, then f(c) is not an extreme value.
    • In reverse, if f(c) is a local extremum, then either f'(c) does not exist(like: f(x) = |x|) or f'(c) = 0.

How do I differentiate e^x? #

• why looking for e? we’re looking for a function that it’s derivative equal to itself: $f(x) = f’(x) = f’’(x)…$. Assuming $f(a) = f’(a) = f’’(a) = … = 1$, how to find a?

• To function $\displaystyle g(x) = \lim_{h \to 0}\frac{x^h-1}{h}$, we know:

  • $\displaystyle f’(2) = \lim_{h \to 0}\frac{2^h-1}{h} \approx 0.693$
  • $\displaystyle f’(3) = \lim_{h \to 0}\frac{3^h-1}{h} \approx 1.099$

• We also know it’s an continuous function, so there much be an a such that $\displaystyle f’(a) = \lim_{h \to 0}\frac{a^h-1}{h} = 1$

• We call the value: $e$, $\displaystyle \lim_{h \to 0}\frac{e^h-1}{h} = 1$

• Now switch the variable to h so we can calculate the derivative of function: $f(x) = e^x$:

  • $$\begin{aligned} f’(x) &= \lim_{h \to 0}\frac{e^{x+h} - e^x}{h} \\ &= \lim_{h \to 0}\frac{e^{x}e^{h} - e^x}{h} \\ &= \lim_{h \to 0}\frac{e^{x}(e^{h} - 1)}{h} \\ &= e^{x} * \lim_{h \to 0}\frac{(e^{h} - 1)}{h} \end{aligned}$$

• Since we already assume that $\displaystyle \lim_{h \to 0}\frac{e^h-1}{h} = 1$, we got: $$f’(x) = e^{x} \cdot 1 = e^{x} = f(x)$$

Review Questions #

• what is the production rule of two derivative functions?
• the quotient rule?
• what is the derivatives of $f(x) = x^3 - x$ and the derivative of the derivative of f(x)?
  • what can the second derivative tell about function f(x)?
  • what is inflection point?
• definition of extreme values, aka local/global minimum/maximum?
  • what is Fermat’s Theorem?
• how to prove the derivative of function$f(x) = e^x$ is itself?