Using the limit definition, how do you find the derivative of f(x) =sqrt (x 1) ?

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= √x  y=x12 Differentiate w.r.t "x" on both sides: dydx=ddx[x12] dydx=12x12−1 (because ddx[xn]=nxn−1) dydx=12x−12 And it can also be written as: dydx=12√x Or, if you meant the limit definition of the derivative function it would look like this: f'(x)=limh→0f(x+h)−f(x)h f'(x)=limh→0√x+h−√xh Now, we multiply the numerator and the denominator by the conjugate of the numerator (conjugates are the sum and difference of the same two terms such as a + b and a - b). f'(x)=limh→0√x+h−√xh⋅√x+h+√x√x+h+√x Since (a+b)(a−b)=a2−b2 we get f'(x)=limh→0x+h−xh(√x+h+√x) Simplifying, we get f'(x)=limh→0hh(√x+h+√x) f'(x)=limh→01√x+h+√x If we evaluate the limit by plugging in 0 for h we get f'(x)=1√x+0+√x=1√x+√x=12√x

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