Expert: Paul Klarreich - 7/25/2009

Question
QUESTION: Hi there;
It has been a LONG time since I have done this stuff.  Can you walk me thru how to do this please?

Find the differential of the function f(x) = 12x^3 - 2x


I have another one also:
If 2xy = 3x+5y^2, find dy/dx by implicit differentiation.


THANK YOU

ANSWER: Questioner: Callie
Country: United States
Category: Calculus
Private: No
Subject: Differential of a function
Question: Hi there;
It has been a LONG time since I have done this stuff.  Can you walk me thru how to do this please?

Find the differential of the function f(x) = 12x^3 - 2x


I have another one also:
If 2xy = 3x+5y^2, find dy/dx by implicit differentiation.


THANK YOU
....................................
The differential is nothing more than  dy = f'(x) dx.

In that case,  dy = (36x^2 - 2) dx

................
When you do I.D., you:

0) Don't solve for x.
1) Differentiate each term, assuming that y is a function of x, so in each place you see a y, you WILL SEE a dy/dx, in some way or another. You might need the product rule (first term in your ex.) and the chain rule (last term in your ex.).
2) Solve algebraically for dy/dx.  The answer may well involve x and y.

2xy = 3x+5y^2

2xy is a product.  D(2xy) = 2y + 2x dy/dx

5y^2 uses the Chain rule:  D(5y^2) = 10y dy/dx

2y + 2x dy/dx = 3 + 10y dy/dx

Solve:

2y - 3 =  10y dy/dx - 2x dy/dx

2y - 3 =  (10y - 2x) dy/dx
        2y - 3  
dy/dx = ----------
        10y - 2x

That's it.


---------- FOLLOW-UP ----------

QUESTION: Thank you so much.  
I came across this reviewing for my midterm next week - Can you provide an example of this concept?

Suppose f and g are both discontinuous at a. Is the sum of f+g necessarily discontinuous at a? Explain.  

Answer
Suppose f and g are both discontinuous at a. Is the sum of f+g necessarily discontinuous at a? Explain.

No.  Try this pair of functions:
      |  1, when  x >= 0
f(x) = |
      | -1, when  x < 0

obviously discontinuous at  x = 0.

And:

g(x) = -f(x), likewise.

Both are discontinuous at  x = 0.  But....

I am sure you could come with other examples.