- #1
The_Brain
- 42
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How do you solve the derivative of x^x? I'm sure it's fairly easy-- I'm just beginning calc though and none of the "forumlas" work.
How can the derivative of x^x be solved?
- Thread starter The_Brain
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In summary, to solve for the derivative of x^x, take the natural log of both sides, differentiate with respect to x, and then use the chain rule and product rule. You can also apply this method to solve for the derivative of y = x^a, where a is a constant. Alternatively, you can use the formula ex*ln(x) to solve for the derivative. The power rule is not applicable in this case.
- #2
StephenPrivitera
- 363
- 0
Take the natural log of both sides.
y=xx
lny=xlnx
Differentiate with respect to x.
(1/y)(dy/dx)=lnx+x/x (chain rule, product rule)
dy/dx=y(lnx+1)
You could do the same thing with y=xa, where a is a constant.
lny=alnx
dy/dx=y(a/x)=xa(a/x)=axa-1
a formula I'm sure you know well
y=xx
lny=xlnx
Differentiate with respect to x.
(1/y)(dy/dx)=lnx+x/x (chain rule, product rule)
dy/dx=y(lnx+1)
You could do the same thing with y=xa, where a is a constant.
lny=alnx
dy/dx=y(a/x)=xa(a/x)=axa-1
a formula I'm sure you know well
Last edited:
- #3
Guybrush Threepwood
- 520
- 1
or xx = ex*ln(x) ad continue from here...
(ex*ln(x))' = ex*ln(x) * (x*ln(x))' and so on...
(ex*ln(x))' = ex*ln(x) * (x*ln(x))' and so on...
- #4
Dx
check out the power rule, learn it , know it, LOVE IT! Its easy, i am just playing but check it out, ok. Any problems with it let me know, k.
Dx
- #5
quantumdude
Staff Emeritus
Science Advisor
Gold Member
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- 24
No, the power rule is not applicable here because the exponent is not a constant. Stephen Privitera's solution is correct; go with that.
FAQ: How can the derivative of x^x be solved?
What is the derivative of x^x?
The derivative of x^x is (x^x)(lnx+1).
How do you solve the derivative of x^x?
To solve the derivative of x^x, use the formula (x^x)(lnx+1) and follow the basic rules of differentiation.
What are the steps to solve the derivative of x^x?
The steps to solve the derivative of x^x are:
- Write the function in the form of f(x)^g(x).
- Take the natural log of the function.
- Use the product rule to find the derivative of the natural log.
- Apply the chain rule to find the derivative of the exponent.
- Multiply the two derivatives together to get the final answer.
What is the purpose of solving the derivative of x^x?
The purpose of solving the derivative of x^x is to find the rate of change of a function at a specific point, which can be useful in optimization problems and in understanding the behavior of a function.
Can the derivative of x^x be simplified?
Yes, the derivative of x^x can be simplified to x^x(lnx+1) by combining like terms and using logarithmic rules.