AboutPaul Klarreich Expertise All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions.
I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.
Experience I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.
Question Find the derivative of 8/√(x-2) using the definition of a derivative.
I have attempted this problem but for some odd reason I can't seem to reach the answer.
WORK:
lim 8/√(x+h-2) - 8/√(x-2)
h->0 h
After this point I don't know what algebra to use to get the answer, I tried conjugate, common denominator, squaring the whole thing.
Answer Questioner: Davis
Country: United States
Category: Calculus
Private: No
Subject: Finding derivative using the definition of a derivative
Question: Find the derivative of 8/√(x-2) using the definition of a derivative.
I have attempted this problem but for some odd reason I can't seem to reach the answer.
WORK:
lim 8/√(x+h-2) - 8/√(x-2)
h->0 ---------------------
h
After this point I don't know what algebra to use to get the answer, I tried conjugate, common denominator, squaring the whole thing.
..................................................
Why don't we start with some lowest common denominator stuff:
LCD = √(x+h-2) √(x-2)
as in let's multiply by it, to clear fractions. After every term gets 'hit' by it:
your original limit:
lim 8/√(x+h-2) - 8/√(x-2)
h->0 ---------------------
h
becomes:
lim 8√(x-2) - 8√(x+h-2)
h->0 --------------------- =
h √(x+h-2) √(x-2)
lim √(x-2) - √(x+h-2)
h->0 8 ------------------
h √(x+h-2) √(x-2)
NOW rationalize (multiply by conjugate) and we take out that '8'.
lim x - 2 - x - h + 2
h->0 8 -------------------------------------
h √(x+h-2) √(x-2) (√(x-2) + √(x+h-2))
lim - h
h->0 8 -------------------------------------
h √(x+h-2) √(x-2) (√(x-2) + √(x+h-2))
Things will finish up nicely now -- I'll leave that to you. Isn't it great when things fall into place?
[Note: These special symbols cause trouble. I hope the square root symbol does not get messed up.]
