sqrt(x^2 - 9) / x
I need to solve this using the definition of a derivative [f(a+h) - f(a)]/h
I can't seem to get it :\
The way to do this is to use a method like one used when working with complex numbers in the denominator.
(f(x+h) - f(x))/h = 1/h * [sqrt((x+h)^2-9)/(x+h) - sqrt(x^2-9)/x]
Put this all over a common denominator:
[x*sqrt((x+h)^2-9) - (x+h)*sqrt(x^2-9)]/hx(x+h)
Now, we want to use the fact that (u+v)(u-v) = u^2-v^2. This will help us eliminate all those pesky square roots. at least in the numerator. SO, multiply top and bottom by
x*sqrt((x+h)^2-9) + (x+h)*sqrt(x^2-9)
Cranking out the math, we now end up with the fraction
9h*(2x+h)/[hx(x+h) * (x*sqrt((x+h)^2-9) + (x+h)*sqrt(x^2-9)]
The lone h top and bottom cancels out as desired, leaving us with:
9*(2x+h)/[x(x+h) * (x*sqrt((x+h)^2-9) + (x+h)*sqrt(x^2-9)]
Now, we can take the limit as h goes to zero, with no pesky 0/0 problems. Just throw them away, giving us
9*(2x)/[x(x) * (x*sqrt((x)^2-9) + (x)*sqrt(x^2-9)]
= 18x/(x^2 * (2x * sqrt(x^2-9))
= 18x/(2x^3 * sqrt(x^2-9)
= 9 / (x^2 * sqrt(x^2-9)
*whew*
Thanks a lot!
To solve the derivative of the function f(x) = sqrt(x^2 - 9) / x using the definition of a derivative [f(a+h) - f(a)]/h, we need to follow a step-by-step process. Here's how you can approach it:
Step 1: Rewrite the function
Rewrite the given function in a simplified form if necessary. Recall that the square root can also be represented as a fractional exponent. In this case, we have:
f(x) = (x^2 - 9)^(1/2) / x
Step 2: Determine f(a+h)
Now, we need to find f(a+h). Substitute (a+h) wherever you see an x in the function:
f(a+h) = [(a+h)^2 - 9]^(1/2) / (a+h)
Step 3: Determine f(a)
Similarly, find f(a) by substituting a into the function:
f(a) = (a^2 - 9)^(1/2) / a
Step 4: Find [f(a+h) - f(a)]/h
Substitute the expressions for f(a+h) and f(a) found in Steps 2 and 3 into the definition of the derivative:
[f(a+h) - f(a)]/h = {[(a+h)^2 - 9]^(1/2) / (a+h) - (a^2 - 9)^(1/2) / a} / h
Step 5: Simplify the expression
Combine the fractions by finding a common denominator, if necessary. Rationalize the denominators if needed, to remove any radical denominators.
Step 6: Take the limit
Finally, take the limit of the expression as h approaches 0. Simplify the expression further if possible.
Note: The process of simplifying this derivative can be quite involved and may require additional algebraic manipulations or trigonometric identities. It's important to double-check and show all the necessary steps to ensure accuracy.