if f(x)= x+(1/x), find f'(x) using the limit definition of derivative.
Please show steps!!!
Thank you!!!
Oh, derivatives, my favorite mathematical circus act! Let's put on a spectacular show and find the derivative of f(x) = x + (1/x) using the limit definition.
To find f'(x), we'll have to use the formula for the derivative. The formula states that the derivative of a function f(x) is equal to the limit as h approaches 0 of [f(x+h) - f(x)] / h.
Step right up, ladies and gentlemen, and let's get started!
f(x) = x + (1/x)
First, let's find f(x+h):
f(x+h) = (x + h) + (1 / (x + h))
Next, we'll subtract f(x) from f(x+h):
f(x+h) - f(x) = [(x + h) + (1 / (x + h))] - (x + (1/x))
Are you ready for the grand finale? Let's divide this by h:
[f(x+h) - f(x)] / h = [(x + h) + (1 / (x + h))] - (x + (1/x)) / h
Now, let's combine those like terms:
[f(x+h) - f(x)] / h = (x + h - x) + (1 / (x + h) - 1/x) / h
Oh, the suspense is building! Now, let's simplify:
[f(x+h) - f(x)] / h = h + (1 / ((x+h)x) - 1/x) / h
And finally, let's take the limit as h approaches 0:
f'(x) = lim(h->0) [h + (1 / ((x+h)x) - 1/x) / h]
Hmm, it seems like this circus act is becoming quite challenging. I'm afraid it can't be simplified any further using the limit definition of the derivative. But don't you worry, my friend, there are other methods that can be used to find the derivative of this function. Keep up the great work and remember to never lose your sense of humor in the world of mathematics!
To find the derivative of the function f(x) = x + (1/x) using the limit definition of derivative, we will follow these steps:
Step 1: Write down the limit definition of the derivative:
f'(x) = lim(h -> 0) [(f(x + h) - f(x))/h]
Step 2: Evaluate f(x + h):
f(x + h) = (x + h) + (1/(x + h))
Step 3: Subtract f(x) from f(x + h):
f(x + h) - f(x) = [(x + h) + (1/(x + h))] - (x + 1/x)
Step 4: Simplify the numerator:
[(x + h) + (1/(x + h))] - (x + 1/x) = [(x + h) + (1/(x + h))] - [(x^2 + 1)/x]
Step 5: Combine like terms:
[(x + h) + (1/(x + h))] - [(x^2 + 1)/x] = [x + h + (1/(x + h)) - (x^2 + 1)/x]
Step 6: Simplify further:
[x + h + (1/(x + h)) - (x^2 + 1)/x] = [x + h + (x - (x^2 + 1))/x(x + h)]
Step 7: Divide by h:
[x + h + (x - (x^2 + 1))/x(x + h)]/h
Step 8: Apply the limit as h approaches 0:
lim(h -> 0) [x + h + (x - (x^2 + 1))/x(x + h)]/h
Step 9: Simplify the expression:
lim(h -> 0) [(2x^2h + (x^2 + 1)h + x(x - (x^2 + 1)))/(xh(x + h))]
Step 10: Cancel out the h in the numerator and denominator:
lim(h -> 0) [(2x^2 + (x^2 + 1) + x(x - (x^2 + 1)))/(x(x + h))]
Step 11: Expand and simplify the numerator:
lim(h -> 0) [(2x^2 + x^2 + 1 + x^2 - x^3 - x + x^3 + x)/(x(x + h))]
Step 12: Combine like terms in the numerator:
lim(h -> 0) [(4x^2 + 1)/(x(x + h))]
Step 13: Simplify the denominator:
lim(h -> 0) [(4x^2 + 1)/(x^2 + hx)]
Step 14: Factor out an h from the denominator:
lim(h -> 0) [(4x^2 + 1)/(h(x + x^2/h))]
Step 15: Cancel out the h from the numerator and denominator:
lim(h -> 0) [(4x^2 + 1)/(x + x^2/h)]
Step 16: Apply the limit as h approaches 0:
(4x^2 + 1)/(x + x^2/0)
Step 17: Since the denominator is 0, the value of the derivative does not exist.
Therefore, the derivative of f(x) = x + (1/x) using the limit definition of derivative is undefined.
To find the derivative of f(x) = x + (1/x) using the limit definition of the derivative, we need to find the expression for the derivative of f(x), which is denoted as f'(x) or df(x)/dx.
The limit definition of the derivative is given by the following formula:
f'(x) = lim(h->0) [(f(x + h) - f(x))/h]
Let's apply this definition to find the derivative of f(x).
1. Start by substituting f(x) = x + (1/x) into the formula:
f'(x) = lim(h->0) [(x + h + (1/(x + h))) - (x + (1/x))]/h
2. Simplify the expression inside the limit:
f'(x) = lim(h->0) [(x + h + ((x - (1/h))/x(x + h))) - (x + (1/x))]/h
= lim(h->0) [(x + h + ((x(x + h) - 1)/(h*x(x + h)))) - (x + (1/x))]/h
= lim(h->0) [(x + h + (x^2 + xh - 1)/(h*x(x + h))) - (x + (1/x))]/h
= lim(h->0) [(x(x + h) + (x^2 + xh - 1))/(h*x(x + h))) - (x + (1/x))]/h
= lim(h->0) [(x^2 + xh + x^2 + xh - 1)/(h*x(x + h))) - (x + (1/x))]/h
= lim(h->0) [(2x^2 + 2xh - 1)/(h*x(x + h))) - (x + (1/x))]/h
3. Combine like terms:
f'(x) = lim(h->0) [(2x^2 + 2xh - 1 - hx - h/x)/(h*x(x + h))] / h
4. Simplify further by canceling out h terms:
f'(x) = lim(h->0) [(2x^2 + xh - hx - 1 - h/x)/(h*x(x + h))]
5. Apply the limit as h approaches 0:
f'(x) = (2x^2 - 1)/x^2
So, the derivative of f(x) = x + (1/x) using the limit definition of the derivative is f'(x) = (2x^2 - 1)/x^2.
f(x+h) = (x + h) + 1/(x+h)
= [(x+h)^2 + 1] /(x+h)
= [x^2 + 2 x h + h^2 + 1 ] / (x+h)
f(x) = (x^2+1)/x
f(x+h) - f(x)
=[x^2+2xh+h^2+1]/(x+h) - (x^2+1)/x
= x^3+2x^2h+h^2x+x -(x^2+1)(x+h)
--------------------------------
x^2 + hx
divide by h
x^3+2x^2h+h^2x+x -(x^2+1)(x+h)
--------------------------------
h x^2 + h^2 x
multiply numerator out
x^3+2x^2h+h^2x+x - x^3 -x -x^2h-h
------------------------------------
h x^2 + h^2 x
x^2 h + h^2 x -h
=----------------
h x^2 + h^2 x
= x^2 + h x -1
---------------
x^2 + h x
let h ---->0
(x^2-1)
---------
x^2
which is actually the right answer believe it or not