How do I find the derivative of f(x) = sqrt(x) without using the power rule?

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Homework Statement



Find the derivative of the function f(x) = sqrt(x).

(Bear in mind that cannot use the power rule or anything like it. I must use limh->0.)

Homework Equations



limh->0 (f(x+h) - f(x))/h

The Attempt at a Solution



I'm getting hung up on expanding out the sqrt(x+h) term. I'm having trouble with exponents less than 1.
 
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Write out the difference quotient and multiply numerator and denominator by sqrt(x-h)+sqrt(x). That's the usual trick.
 
Yea, you're definitely right. Thanks! Been a while since I had to do that.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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