Question

Find the derivative of f(x)=1/sqrt(x) using the DEFINITION OF THE DERIVATIVE. (please show ALL steps)

Answer 1

t(x+h) = 1/√(x+h); [f(x+h) - f(x)]/h = [1/√(x+h)] -1/√x]/h = [√x - √(x+h)]/[h√x(x+h)]
Now multiply numerator and denominator by √x + √(x+h)

Let me work them separately as the algebra gets messy:

Numerator: [√x - √(x+h)][√x + √(x+h)] = x - x - h = -h and this is why we multiply by √x + √(x+h): the -h in the numerator will cancel the h in the denominator

Denominator:

[h√x(x+h)][√x + √(x+h)] = h[x√(x+h) +(x+h)√x]

Numerator/Denominator: -h/{h[x√(x+h) +(x+h)√x] } = -1/[x√(x+h) +(x+h)√x]
Let h ->0 and get -1/(2x√x) = -1/(2x^3/2)

Answer 2

http://answers.yahoo.com/question/index?qid=20110914200801AA9TAaE

Answer 3

thanks