# Derivative of Square Root of x at 0

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• mopit_011
In summary, the power rule and the definition of the derivative yield different results for the derivative of the square root at x = 0. While the power rule gives an undefined result, the definition of the derivative yields infinity. This is due to the fact that the derivative of the square root is undefined at x = 0 in both cases.
mopit_011
When you use the power rule to differentiate the square root, the result is 1/2(sqrt. x) which is undefined at 0. But, when you use the definition of the definition of the derivative to calculate it, the result is infinity. What causes this difference between these two methods?

Infinity is undefined in this context.

mopit_011 said:
When you use the power rule to differentiate the square root, the result is 1/2(sqrt. x) which is undefined at 0. But, when you use the definition of the definition of the derivative to calculate it, the result is infinity. What causes this difference between these two methods?
None. The derivative of ##x\longmapsto \sqrt{x}## isn't defined for ##x=0## in neither case.

Some might also observe that
$$\lim_{x\to 0^+} \frac{1}{2\sqrt{x}}$$
Is also infinity.

Infrared

## 1. What is the derivative of the square root of x at 0?

The derivative of the square root of x at 0 is equal to 1/2.

## 2. How is the derivative of the square root of x at 0 calculated?

The derivative of the square root of x at 0 is calculated using the power rule for derivatives, which states that the derivative of x^n is equal to n*x^(n-1). In this case, n=1/2, so the derivative is equal to (1/2)*x^(-1/2), which simplifies to 1/2x^(-1/2) or 1/2x^(1/2).

## 3. Why is the derivative of the square root of x at 0 equal to 1/2?

This can be understood by graphing the square root function and observing that at x=0, the slope of the tangent line is equal to 1/2. Additionally, the power rule for derivatives can be applied to find the derivative of the square root function at 0.

## 4. Is the derivative of the square root of x continuous at 0?

Yes, the derivative of the square root of x is continuous at 0. This can be seen by graphing the function and observing that the slope of the tangent line is consistent as x approaches 0 from both the positive and negative sides.

## 5. Can the derivative of the square root of x at 0 be simplified further?

Yes, the derivative of the square root of x at 0, which is 1/2x^(1/2), can be further simplified to 1/2√x. This can be done by factoring out the x^(1/2) term and simplifying the resulting expression.

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