Partial derivatives

Partial derivatives

Lessons

Notes:

Introduction to Partial Derivatives
Since we are dealing with multi-variable functions, we want to take the derivative in respect to 1 variable. This is known as the partial derivative.

For a function f(x,y)f(x,y), we can have two partial derivatives:
  • fx=dfdx→f_x = \frac{df}{dx} \to derivative in terms of xx
  • fy=dfdy→f_y = \frac{df}{dy} \to derivative in terms of yy
When taking partial derivatives of functions in terms of a variable, we treat other variables as constants. In other words, we allow one variable to vary, and the other variables to be fixed.

Definition of Partial Derivatives
Recall from the definition of derivative, we have the formula:

lim⁡h→0f(x+h)−f(x)h\lim\limits_{h \to 0} \frac{f(x+h)-f(x)}{h}

From for the definition of partial derivatives, we have the two following equations:

fx=lim⁡h→0f(x+h,y)−f(x,y)h f_x = \lim\limits_{h \to 0} \frac{f(x+h,y) - f(x,y)}{h}
fy=lim⁡h→0f(x,y+h)−f(x,y)h f_y = \lim\limits_{h \to 0} \frac{f(x, y+h) - f(x,y)}{h}

We won't be using these equations because we already know how to take derivatives.

Application of Partial Derivatives
We can use partial derivatives to find the tangent slope of the traces at a certain point (a,b)(a,b). We can do this by finding fx(a,b)f_x(a,b) and fy(a,b)f_y(a,b).
We can also use partial derivatives to see if f(x,y)f(x,y) is increasing or decreasing. In other words, if

fx>0f_x\gt0, then f(x,y)f(x,y) is increasing as we vary xx.
fx<0f_x\lt0, then f(x,y)f(x,y) is decreasing as we vary xx.
fy>0f_y\gt0, then f(x,y)f(x,y) is increasing as we vary yy.
fy<0f_y\lt0, then f(x,y)f(x,y) is decreasing as we vary yy.
  • Introduction
    Partial Derivatives Overview:
    a)
    Introduction to Partial Derivatives
    • Derivatives in terms of 1 variable
    • Treating all other variables as constants
    • An example

    b)
    Definition of Partial Derivatives
    • Recalling the definition of derivative
    • Two formal equations
    • Won't be Using them (Too Complicated)

    c)
    Application of Partial Derivatives
    • Finding the tangent slope of a trace
    • Seeing if the function is increasing or decreasing

  • 1.
    Finding the Partial Derivatives
    Find the first order partial derivatives of the following function:

    f(x,y)=2xln⁡(xy2)+xy−x3f(x,y) = 2x \ln (xy^2) + \frac{x}{y} - \sqrt{x^3}

  • 2.
    Find the first order partial derivatives of the following function:

    h(s,t)=h(s,t) = sin⁡(es2t3)+tan⁡2t \sin(e^{s^2t^3}) + \tan \frac{2}{t}

  • 3.
    Find the first order partial derivatives of the following function:

    g(r,s)=rln⁡r2+s2+rs g(r,s) = r \ln \sqrt{r^2 + s^2 + rs}

  • 4.
    Find the slope of the traces to z=4−x2−y2 z= \sqrt{4-x^2-y^2} at the point (1,2) (1, \sqrt{2} ) .
  • 5.
    Find the slope of the traces to z=z = sin⁡(xy)\sin(xy) at the point (0,π2)(0, \frac{\pi}{2}).
  • 6.
    Is the Function Increasing or Decreasing?
    Determine if f(x,y)=y2cos⁡(xy)f(x,y)=y^2 \cos (\frac{x}{y}) is increasing or decreasing at the point (π2,1)(\frac{\pi}{2},1) if:
    a)
    We allow xx to vary and hold yy fixed.

    b)
    We allow yy to vary and hold xx fixed.

  • 7.
    Determine if f(x,y)=x2+y2+ln⁡(xy)f(x,y)=x^2+y^2+ \ln (xy) is increasing or decreasing at the point (1,2)(1, 2) if:
    a)
    We allow xx to vary and hold yy fixed.

    b)
    We allow yy to vary and hold xx fixed.

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3.
Partial Derivatives
3.1
Functions of several variables
3.2
Limits and continuity of multivariable functions
3.3
Partial derivatives
3.4
Higher order partial derivatives
3.5
Differentials of multivariable functions
3.6
Chain rule for multivariable functions
3.7
Directional derivatives
Calculus 3