ADVANCED DIFFERENTIATION TECHNIQUES

A)  Implicit differentiation

      1)  Take the derivative of both sides of the equation
      2)  Solve for the derivative expression

B)  Logarithmic differentiation

      1)  Take the natural log of both sides of the equation
      2)  Simplify completely using log properties
      3)  Proceed with implicit differentiation (see above)

Example of A)    Find the slope of the tangent line to the curve  y³ ln(x) = 1 at (e,1)

Solution:   take the derivative of both sides with respect to x.
               
  d[y³ ln(x)] = d(1)   gives y³ (1/x) + [ln(x)] 3y² dy/dx  =  0  (use product rule on left side)
  dx                dx                                   

solve for dy/dx: [ln(x)] 3y² dy/dx =   - y³ (1/x)  gives dy/dx =  - y³ (1/x)/ [ln(x)] 3y²

evaluate dy/dx at x=e and y=1:   dy/dx =  -e/3   (note:  lne=1)


Example of B)   Find dy/dx:    y= x² e^x2 (5x+1)⁵

Solution:   take the natural log of both sides and simplify completely

    lny = ln(x² ) + ln(e^x2 ) + ln[(5x+1)⁵    (use the log of a product property)

    lny = 2lnx +x² lne + 5 ln(5x+1)  (use the "knock-down" power property of logs)

   Only now are you ready for implicit differentiation

    (1/y) dy/dx = 2(1/x) +  2x + 5[1/(5x+1)](5)  (note:  lne=1)

             dy/dx = y[(2/x) +  2x +25/(5x+1)]