Use f(x)= 1/(square root(x)+1) to answer the following;
-- Use the difference quotient f'(x)=lim z->x (f(x)-f(z))/ (x-z) to find f'(x)!
Please show steps!!!
Thank you!!!
f(x) = 1/[ x^.5 +1 ]
f(z) = 1/[ z^.5 + 1 }
f(x)-f(z) = 1/[ x^.5 +1 ] - 1/[ z^.5 + 1]
= ([z^.5+1]-[x^.5+1])/(x^.5z^.5 +x^.5+z^.5+1)
= (z^.5-x^.5) /(x^.5z^.5 +x^.5+z^.5+1)
Divide by (x-z) which is (x^.5-z^.5)(x^.5+z^.5)
and get
-(x^.5+z^.5) / (x^.5z^.5 +x^.5+z^.5+1)
let z --->x
- 2 x^.5 / (x +2 sqrt x + 1)
check my arithmetic !!!
To find the derivative of the function f(x) = 1/√(x+1) using the difference quotient, follow these steps:
1. Start with the difference quotient formula: f'(x) = lim (z→x) [(f(x) - f(z)) / (x - z)]
2. Substitute the given function f(x) = 1/√(x+1) into the formula: f'(x) = lim (z→x) [(1/√(x+1) - 1/√(z+1)) / (x - z)]
3. Simplify the expression inside the limit as much as possible. To do this, combine the fractions by finding a common denominator. The common denominator here is √(x+1) * √(z+1): f'(x) = lim (z→x) [((√(z+1) - √(x+1)) / (√(x+1) * √(z+1))) / (x - z)]
4. Combine the fractions in the numerator: f'(x) = lim (z→x) [(√(z+1) - √(x+1)) / ((x - z) * √(x+1) * √(z+1))]
5. Multiply the conjugate of the numerator to simplify further. The conjugate of √(z+1) - √(x+1) is √(z+1) + √(x+1): f'(x) = lim (z→x) [((√(z+1) - √(x+1)) * (√(z+1) + √(x+1))) / ((x - z) * √(x+1) * √(z+1))]
6. Simplify the numerator using the difference of squares: f'(x) = lim (z→x) [(z+1 - (x+1)) / ((x - z) * √(x+1) * √(z+1))]
7. Simplify further: f'(x) = lim (z→x) [(z - x) / ((x - z) * √(x+1) * √(z+1))]
8. Cancel out the (x - z) from the numerator and denominator: f'(x) = lim (z→x) [-1 / (√(x+1) * √(z+1))]
9. Take the limit as z approaches x: f'(x) = -1 / (√(x+1) * √(x+1))
10. Simplify further: f'(x) = -1 / (x+1)
Thus, the derivative of the function f(x) = 1/√(x+1) is f'(x) = -1 / (x+1).