derivative at a number

Question

Find the derivative of f(x) = sqrt(36-x^2) / 9x when x=2

Answer 1

The thing that makes this problem painful is that there is a lot going on.  The thing that makes it not so bad, is doing one small step at a time. 

You have a quotient, so you will need quotient rule.

You have a radical, and you can use laws of exponents to make that not so bad.

 

Let's start with the quotient rule:

The quotient rule for f(x)=g(x)/h(x)
f'(x)= (g'(x)h(x)-h'(x)g(x))/[h(x)]² It looks pretty ugly without math type.  I am going to use that old third grade symbol for division to make the numerator more distinct from the denominator.

 

f'(x)= (g'(x)h(x)-h'(x)g(x))  ÷  [h(x)]² There less ugly.

 

Let g(x) = √(36-x²)   that is the same as (36-x²)^(1/2)   But the second form is easier to visualize the use of chain rule.

g'(x) = (1/2)(36-x²)^(-1/2)(-2x)  or      -2x/2√(36-x²)  =  x/√(36-x²)
                                                               

Let h(x) = 9x, then h'(x)=9 , and [h(x)]²=81x²

 

Now that you have the parts hashed out.  Substitute them carefully, yet fearlessly into the quotient rule.   Since you a compound rational, feel free to use the ÷ and ignore the denominator for a bit.  It will wait patiently!

 

Given:

f'(x)= (g'(x)h(x)-h'(x)g(x)) ÷ [h(x)]²

 

      =[    -x·/√(36-x²) · 9x  - 9 √(36-x²)]÷ 81x²       Substitution

         

    Factor out and cancel the 9's
  We need a LCD, the least common denominator is √(36-x²)

                           

 

f'(x)=[ -x²/ √(36-x²) - √(36-x²)  · √(36-x²)/√(36-x²)]÷ 9x² 
                      

 

=    -x²- (36-x²) / √(36-x2)   = (-x² - 36 + x²)/(9x^2√(36-x^2) )  = -36/ 9x²√(36-x²)

    = -4 / x²√(36-x²)

                               

 

so f'(x) = -4/(x²√(36-x²),    just substitute 2 in to find f'(2).  And you will be done.

I hope that was helpful.

 

Contact me if you need more help. 

  

 

Answer 2

f'(x) = 0.5sqrt(1/(36-x^2))*(-2x)/(9x) - sqrt(36-x^2) / (9x^2) 

 

f'(2) = -sqrt(2)/8

 

look also here: http://www.derivative-calculator.net/.

 

This site gives detailed description of how to solve the problem