Troubleshooting Differentiation Using the Long Formula

In summary, the conversation was about someone asking for help with differentiating using the long formula. The solution involved creating a common denominator, simplifying the numerator, and taking the limit as h approached 0 to cancel out the remaining h term. The final solution was -1/16.
  • #1
theBTMANIAC
6
0

Homework Statement

I have to differentiate using the long formula [f(x+h)-f(x)] / (h).


Homework Equations



f(x) = (1)/(2+x), x=2

The Attempt at a Solution



First I wrote [ (1/2+x+h) - (1/2+x) ] / (h). Then I created a common denominator in the numerator. I then made the bottom denominator a fraction.

Code:
(2+x)-(2+x+h)     (1)
___________    x  __
(2+x+h)(2+x)      (h)

I subtracted common variables from the numerator and got.

Code:
   -(h)                (1)
________             x ___ 
[(2+x+h)(2+x)]         (h)

Then I got:

       -(1)
_______________
(2+x+h)(2+x)

But as I expanded the numerator, I found that I couldn't get rid of the last h. What is the problem?

Thank you.
 
Last edited:
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  • #2
i think you dropped a minus sign

also remember h is tending to zero... so multiple out, then take the limit & you can cancel terms tending to zero
 
  • #3
note you can write in tex (click on below to see the code)
[tex]\frac{-1}{(2+x)(2+x+h)} = \frac{-1}{(2+x)^2+h(2+x))} [/tex]
 
  • #4
Thanks for the quick reply.

That would mean the solution to this particular problem is [tex]- \frac{1}{16}?[/tex]
 
  • #5
Yes.
 

What is the formula for differentiating 1 over 2+x?

The formula for differentiating 1 over 2+x is -1/(2+x)^2.

What is the process for differentiating 1 over 2+x?

The process for differentiating 1 over 2+x is to use the power rule, where you bring the exponent down in front and subtract 1 from the original exponent. In this case, the original exponent is -1, so it becomes -2 after differentiation. The denominator remains the same, and the negative sign in front indicates that the derivative is negative.

Why is the derivative of 1 over 2+x negative?

The derivative of 1 over 2+x is negative because the original function is a decreasing function. As x increases, the value of the function decreases.

Can the formula for differentiating 1 over 2+x be simplified?

Yes, the formula for differentiating 1 over 2+x can be simplified to -1/(x+2)^2. This is the same as the original formula, but with the denominator in a simpler form.

What is the significance of differentiating 1 over 2+x in scientific applications?

Differentiating 1 over 2+x can be used in various scientific applications, such as in physics to find the velocity of an object with changing acceleration or in chemistry to determine the rate of a chemical reaction. It is an important tool in analyzing and understanding the behavior of functions and their derivatives in various scientific fields.

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