Find the derivative of the function by the limit process.
f(x)=1/sqrt(x)
I've worked through my problem and got f'(x) to be -1/2x but the actual answer is -1/2x^3/2? Where did the 3/2 power come from?
Any help is greatly appreciated!
f(x+h)-f(x) = 1/√(x+h) - 1/√x
= (√x - √(x+h))/√(x(x+h))
= (x-(x+h))/(√x(x+h))(√x+√(x+h))
= -h/(√x(x+h))(√x+√(x+h))
divide by h and you have
-1/(√x(x+h))(√x+√(x+h))
as h->0, that is
-1/(√x^2 (2√x))
= -1/(x*2√x)
I suspect a √ got lost somewhere in the shuffle. Should have shown your work. I might have picked it up more easily than doing it myself...
To find the derivative of the function f(x) = 1/sqrt(x) using the limit process, we can follow these steps:
Step 1: Start by writing down the difference quotient, which is the definition of the derivative using the limit process. The difference quotient for f(x) is given by:
(f(x + Δx) - f(x))/Δx
Step 2: Substitute the function f(x) into the difference quotient:
[1/sqrt(x + Δx) - 1/sqrt(x)]/Δx
Step 3: Simplify the expression using a common denominator. Multiply the numerator and denominator by sqrt(x + Δx) and sqrt(x), respectively:
[(sqrt(x) - sqrt(x + Δx))/(sqrt(x + Δx) * sqrt(x))]/Δx
Step 4: Combine the fractions in the numerator by finding a common denominator:
[(sqrt(x) - sqrt(x + Δx))/(sqrt(x + Δx) * sqrt(x))]/Δx * [(sqrt(x) + sqrt(x + Δx))/(sqrt(x) + sqrt(x + Δx))]
Step 5: Simplify the numerator:
[(sqrt(x) - sqrt(x + Δx))*(sqrt(x) + sqrt(x + Δx))]/[(sqrt(x + Δx) * sqrt(x))(sqrt(x) + sqrt(x + Δx))]/Δx
Step 6: Cancel out the common factors in the denominator:
[(sqrt(x) - sqrt(x + Δx))*(sqrt(x) + sqrt(x + Δx))]/[(sqrt(x + Δx) * sqrt(x))(sqrt(x) + sqrt(x + Δx))]/Δx
Step 7: Expand the numerator:
[(sqrt(x) - sqrt(x + Δx))*(sqrt(x) + sqrt(x + Δx))]/[(sqrt(x + Δx) * sqrt(x))(sqrt(x) + sqrt(x + Δx))]/Δx
= [(sqrt(x) * sqrt(x) + sqrt(x)*sqrt(x + Δx) - sqrt(x + Δx)*sqrt(x) - sqrt(x + Δx)*sqrt(x + Δx))]/[(sqrt(x + Δx) * sqrt(x))(sqrt(x) + sqrt(x + Δx))]/Δx
Step 8: Simplify the numerator further:
= [sqrt(x)^2 - sqrt(x + Δx)*sqrt(x) - sqrt(x + Δx)*sqrt(x) - sqrt(x + Δx)^2]/[(sqrt(x + Δx) * sqrt(x))(sqrt(x) + sqrt(x + Δx))]/Δx
= [x - 2*sqrt(x)*sqrt(x + Δx) - (x + Δx)]/[(sqrt(x + Δx) * sqrt(x))(sqrt(x) + sqrt(x + Δx))]/Δx
= [x - 2*sqrt(x)*sqrt(x + Δx) - x - Δx]/[(sqrt(x + Δx) * sqrt(x))(sqrt(x) + sqrt(x + Δx))]/Δx
= [-2*sqrt(x)*sqrt(x + Δx) - Δx]/[(sqrt(x + Δx) * sqrt(x))(sqrt(x) + sqrt(x + Δx))]/Δx
Step 9: Simplify and cancel out Δx in the numerator:
= [-2*sqrt(x)*sqrt(x + Δx) - Δx]/[(sqrt(x + Δx) * sqrt(x))(sqrt(x) + sqrt(x + Δx))]/Δx
= [-2*sqrt(x)*sqrt(x + Δx)]/[(sqrt(x + Δx) * sqrt(x))(sqrt(x) + sqrt(x + Δx))]
Step 10: Take the limit as Δx approaches 0:
lim(Δx→0)[-2*sqrt(x)*sqrt(x + Δx)]/[(sqrt(x + Δx) * sqrt(x))(sqrt(x) + sqrt(x + Δx))]
Step 11: Simplify the expression by canceling out common factors of Δx:
lim(Δx→0)[-2*sqrt(x)]/[(sqrt(x + Δx))(sqrt(x) + sqrt(x + Δx))]
Step 12: Substitute Δx = 0 into the expression:
lim(Δx→0)[-2*sqrt(x)]/[(sqrt(x))(sqrt(x) + sqrt(x))]
= lim(Δx→0)[-2*sqrt(x)]/[2*sqrt(x)^2]
= lim(Δx→0)[-2*sqrt(x)]/[2*x]
= lim(Δx→0)[-sqrt(x)/x]
Step 13: Finally, evaluate the limit:
= -sqrt(x)/x
Therefore, the derivative of f(x) = 1/sqrt(x) by the limit process is -sqrt(x)/x.
The discrepancy you mentioned in your calculation likely comes from missing a step while simplifying the terms. To get the correct result, make sure to follow each step carefully, simplifying expressions and canceling terms with caution.