Derivative of Square Root

In summary, to find the derivative of (sq. rt of x)^x, you can use the chain rule and apply the concept of a function within a function. The resulting derivative is (1 + ln(x))/2 * x^(x/2).
  • #1
ssmh
1
0
I can't figure out
Derivative of (sq. rt of x)^x

Can anyone help me?
 
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  • #2
Welcome to PF =]

Use the chain rule, this is simply a function within another function. the outer function just happens to be an exponential. [tex]\frac{d}{dx}f^x=f^x ln(f)[/tex] (if I remember correctly)
 
  • #3
if this is what you mean :

find f(x) prime, where f(x) = sqrt(x)^x

Before I help you, you need to help me. Tell me what concept you have learned that could
apply to this problem
 
  • #4
djeitnstine said:
Welcome to PF =]

Use the chain rule, this is simply a function within another function. the outer function just happens to be an exponential. [tex]\frac{d}{dx}f^x=f^x ln(f)[/tex] (if I remember correctly)

No, that's incorrect.
 
  • #5
Here's how I did it.

[tex]\frac{d}{dx}[/tex](xx/2)

[tex]\frac{d}{dx}[/tex]e(x/2)ln(x)

(1/2)(1 + ln(x))e(xln(x))/2

[tex]\frac{1 + ln(x)}{2}[/tex] * x(x/2)
 
  • #6
looks good to me.
 

1. What is the derivative of the square root of x?

The derivative of the square root of x is equal to 1 over 2 times the square root of x. This can be written as 1/(2√x) or (x^1/2)/2.

2. How do you find the derivative of a function involving a square root?

To find the derivative of a function involving a square root, you can use the power rule and the chain rule. First, rewrite the function as (x^1/2) and then apply the power rule to get (1/2)x^(-1/2). Next, use the chain rule by multiplying the derivative by the derivative of what's inside the square root (1/2x). This results in a final derivative of 1/(2√x).

3. Can the derivative of the square root of x be negative?

Yes, the derivative of the square root of x can be negative. This means that the function is decreasing at that point. However, the value of x must be positive in order for the square root to be defined.

4. What is the derivative of the square root of a constant?

The derivative of the square root of a constant is equal to 0. This is because a constant value does not change with respect to x, so its derivative is always 0.

5. Is the derivative of the square root of x the same as the derivative of x^(1/2)?

Yes, the derivative of the square root of x is the same as the derivative of x^(1/2). This is because they are two different ways of representing the same function, with the exponent of 1/2 indicating the square root. Therefore, the derivative will be the same for both representations.

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