using first principles to differentiate y = 1/(√ x).
I tried using this method many times but I get different answers all the time. I can do it using the power rule but I am not sure what I am doing wrong....
my answer is -1/(2x^3/2) using power rule
so why didn't you show your work?
f(x+h)-f(x) = 1/√(x+h) - 1/√x
Now divide by h and you have
(1/√(x+h) - 1/√x)/h
= (√x - √(x+h)) / (h(√x)(√x+h))
= (√x - √(x+h))(√x + √(x+h)) / (h √x √(x+h))
= (x - (x+h)) / (h √x √(x+h) * (√x + √(x+h)))
= -h / (h √x √(x+h) * (√x + √(x+h)))
= -1/ (√x √(x+h) * (√x + √(x+h)))
Now take the limit as h→0 and you have
-1/(√x √x * 2√x)
= -1/(2x^(3/2))
y = 1/(√ x) = x^-0.5
dy/ dx = -0.5 x^-1.5 so I agree with your answer
now
y ( x+h )= (x+h)^-.5 = 1/(x+h)^.5
y (x) = 1/x^.5
y (x+h) -y(x) = 1/(x+h)^.5 - 1/x^.5
= [ x^.5 - (x+h)^.5] / [x^.5 (x+h)^.5]
we want that divided by h as h --->0
well (a+b)^.5 = a^.5 + ,5 a^-.5 b +.5 *.5*-.5*a^-1.5 bb^2 .... series
so [ x^.5 - x^-5 - .5 x^-.5 h .....] / h [x^.5 (x+h)^.5]
let h ---> 0
-.5 x^-.5 / x
= -.5 x^-1.5
Caramba
To differentiate the function y = 1/(√ x) using first principles, we'll start from the definition of the derivative:
lim(h -> 0) [f(x + h) - f(x)] / h
First, let's find f(x + h):
f(x + h) = 1 / (√(x + h))
Next, substitute these values into the derivative definition:
lim(h -> 0) [1 / (√(x + h)) - 1 / (√x)] / h
To simplify the expression, we'll use a technique called "rationalizing the denominator". Multiply the numerator and denominator by the conjugate of the numerator:
lim(h -> 0) [(1 / (√(x + h)) - 1 / (√x)) * (√(x + h)) / (√(x + h))] / h
Simplifying this further:
lim(h -> 0) [(√(x + h) - √x) / (√((x + h) * x))] / h
Now, let's simplify the numerator by multiplying it with its conjugate:
lim(h -> 0) [(√(x + h) - √x) * (√(x + h) + √x) / [h * (√((x + h) * x)) * (√(x + h) + √x)]]
Expanding the numerator:
lim(h -> 0) [(x + h - x) / [h * (√((x + h) * x)) * (√(x + h) + √x)]]
Simplifying further:
lim(h -> 0) [h / [h * (√((x + h) * x)) * (√(x + h) + √x)]]
Canceling out the h terms:
lim(h -> 0) 1 / [√((x + h) * x) * (√(x + h) + √x)]
As h approaches 0, the expression becomes:
1 / [√(x * x) * (√x + √x)]
Now, simplify further:
1 / [2x * 2√x]
1 / [4x√x]
Therefore, using first principles, the derivative of y = 1/(√ x) is 1 / [4x√x].
It's important to note that your answer using the power rule, -1/(2x^3/2), is incorrect. The power rule is applicable for differentiating functions of the form f(x) = x^n, but it does not work for this particular function.