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canucks81
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Homework Statement
Use the definition of derivative to Find f ' (2) if f(x) = 1/(x+2)^2
Homework Equations
lim [ f(a+h) - f(a) ]/ h
h->0
3. Attempt
The answer I get is 1/8. Can someone tell me if this is correct? Thanks
canucks81 said:Homework Statement
Use the definition of derivative to Find f ' (2) if f(x) = 1/(x+2)^2
The answer I get is 1/8. Can someone tell me if this is correct? Thanks
No, it's not.canucks81 said:lim x-> 2 (1/(x+2)^2) - 2 / (x-2)
lim x ->2 [ 1 - 2(x+2)^2 / (x+2)^2 ] / (x-2)
lim x->2 [ 1 - (2x^2 + 8x + 8 ) / (x+2)^2 ] / (x-2)
lim x->2 [ 1 -(2x+4)(x+2) / (x+2)^2 ] / (x-2)
lim x ->2 [ 1 - 2 (x+2)^2 / (x+2)^2 ] / (x-2 )
(x+2)^2 cancel out
lim x -> 2 -1/(x-2) = -1 / (2-2) = -1/0 = negative infinity
My mistake was that I used (x+2) instead of (x+2)^2 so I was just a little careless. I'm pretty sure this is the correct answer.
canucks81 said:I still feel like my 3rd step is wrong. If i write it out as 1 - 2x^2 - 8x - 8 then this will eventually turn into -2x^2 - 8x - 7.
You started out wrong right off the bat. This is not the correct difference quotient for f(x) = 1/(x + 2)2. See my previous post for a better way to start.canucks81 said:lim x-> 2 (1/(x+2)^2) - 2 / (x-2)
No.canucks81 said:so using [f(a+h) - f(a) ] / h
is this the correct set up? [(1/(2+h)^2 - 1/16)] (1/h)
The formula for finding the derivative of a function at a specific point is f'(x) = lim(h->0) [f(x+h) - f(x)]/h. For this problem, we substitute x=2 into the given function to find f'(2).
To find the derivative of a fraction, we use the power rule of differentiation. In this case, we have a fraction raised to the power of -2, so the derivative becomes -2(x+2)^-3. We then substitute x=2 into this expression to find f'(2).
f'(2) represents the slope of the tangent line to the function f(x) at the point x=2. This is also known as the instantaneous rate of change of the function at x=2.
Finding f'(2) allows us to understand the behavior of the function at the specific point x=2. It can help us determine if the function is increasing or decreasing at that point and the steepness of the curve at that point.
Yes, there is a shortcut called the power rule of differentiation. This rule states that the derivative of x^n is nx^(n-1), where n is any real number. In this case, we use the power rule to find the derivative of our given function, and then substitute x=2 into the resulting expression to find f'(2).