Expert: Paul Klarreich - 2/26/2007

Question
1.) x = 1/2 t-to the fourth power - 5t - 3
2.) y = 1 all over square root of x-2

Answer
Questioner:   Julie
Category:  Calculus
 
Subject:  First and Second Derivative Function
Question:  1.) x = 1/2 t-to the fourth power - 5t - 3
2.) y = 1 all over square root of x-2
.............................................
Hi, Julie,

You didn't parenthesize much, so I am guessing about what you wrote.  And please include the instructions with each example.  I will guess from your subject line, but it's much better if you include them -- I might guess wrong.

1.  x = 1/2(t^4 - 5t - 3)

x' = 1/2(4t^3 - 5)

x'' = 1/2(12t^2) = 6t^2
         1
2. y = --------- = (x-2)^-1/2
      sqrt(x-2)

y' = (-1/2)(x-2)^-3/2

Note 1: this is a chain rule example, but u = x-2, and du/dx = 1

Note 2: When you apply the  x^n rule, giving  nx^n-1, your 'n' is -1/2, and -1/2 - 1

= -3/2
       -1
y' = ----------
    2(x-2)^3/2     

or

         -1
y' = ---------------
    2(x-2)sqrt(x-2)     

Now, for y'', go back to the exponent form:

y' = (-1/2)(x-2)^-3/2

y'' = (-1/2)(-3/2)(x-2)^-5/2
          3
y'' = -------------
     4(x - 2)^5/2