Week 5 - Techniques of Differentiation

1. The Product Rule

  • If f and g are both differentiable, then ddx[f(x)g(x)]=f(x)ddxg(x)+ddxf(x)g(x)\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 Δx\Delta x, then the corresponding changes in u and v are
        • Δu=f(x+Δx)f(x)\Delta u = f(x + \Delta x) - f(x), Δv=g(x+Δx)g(x)\Delta v = g(x + \Delta x) - g(x)
      • The change in the area of the rectangle is:
        • Δ(uv)=(u+Δu)(v+Δv)uv=uΔv+vΔu+ΔuΔv\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 Δx\Delta x, we get Δ(uv)Δx=uΔvΔx+vΔuΔx+ΔuΔvΔx\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 Δx0\Delta x \to 0, we get the derivative of uv(f(x)g(x)):

        ddx(uv)=limΔx0Δ(uv)Δx=limΔx0(uΔvΔx+vΔuΔx+ΔuΔvΔx)=ulimΔx0ΔvΔx+vlimΔx0ΔuΔx+(limΔx0Δu)(limΔx0(ΔvΔx)=uddxv+vddxu+0uddxv\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 * u\frac{d}{dx}v \end{aligned}

      • ddx(uv)=uddxv+vddxu\frac{d}{dx}(uv) = u\frac{d}{dx}v + v\frac{d}{dx}u

    • use limit theorem

      ddx(f(x)g(x))=limh0f(x+h)g(x+h)f(x)g(x)h=limh0f(x+h)g(x+h)f(x+h)g(x)+f(x+h)g(x)f(x)g(x)h=limh0f(x+h)g(x+h)f(x+h)g(x)h+limh0f(x+h)g(x)f(x)g(x)h=limh0g(x+h)g(x)hlimh0f(x+h)+limh0f(x+h)f(x)hlimh0g(x)=ddxg(x)limh0f(x+h)+ddxf(x)limh0g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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}*\lim_{h \to 0}f(x+h) + \lim_{h \to 0}\frac{f(x+h)-f(x)}{h}*\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: (fg)=fg+gf(f \cdot g)'=f \cdot g'+g \cdot f'

2. The Quotient Rule

  • If f and g are differentiable, then

    • ddx[f(x)g(x)]=ddxf(x)g(x)f(x)ddxg(x)g(x)2\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: (fg)=gffgg2(\frac{f}{g})'=\frac{g \cdot f'-f \cdot g'}{g^2}

3. 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)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)=x3xf(x) = x^3 - x
    • f(x)=3x21f'(x)=3x^2-1, f(x)=6xf''(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)=dsdtv(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)a(t)=v'(t)=s''(t)
      • or in Leibniz notation: a=dvdt=d2sdt2a=\frac{dv}{dt}=\frac{d^2s}{dt^2}

3.1. 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.

4. Extreme Value

4.1. 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).

4.2. Fermat’s Theorem (Find Extreme Value)

  • Suppose f is a function, defined on the interval (a,b)c(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.

4.3. e^x

  • To function g(x)=limh0xh1hg(x) = \lim_{h \to 0}\frac{x^h-1}{h}, we know:
    • limh02h1h0.693\lim_{h \to 0}\frac{2^h-1}{h} \approx 0.693
    • limh03h1h1.099\lim_{h \to 0}\frac{3^h-1}{h} \approx 1.099
  • Then there is an x such that limh0xh1h=1\lim_{h \to 0}\frac{x^h-1}{h} = 1.
  • We call the x value: ee, limh0eh1h=1\lim_{h \to 0}\frac{e^h-1}{h} = 1
  • To calculate the derivative of exe^x: f(x)=limh0ex+hexh=limh0exehexh=limh0ex(eh1)h=exlimh0(eh1)h\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}

  • We already assume that limh0eh1h=1\lim_{h \to 0}\frac{e^h-1}{h} = 1, so we got: f(x)=ex1=ex=f(x)f'(x) = e^{x} * 1 = e^{x} = f(x)

results matching ""

    No results matching ""