How to derive a number to the power x (e.g. 2^x)

[develish16]

i recently started my calculus III course but i was stuck with the derivative part when i derive a real number to the power x ( like 2^x) and also when deriving x^x. does anybody have an idea or a link in which these kinds of derivatives can be solved. thanks for ur help

 

[HallsofIvy]

Use "logarithmic differentiation". If $y= x^x$, then $ln(y)= ln(x^x)= xln(x)$. Use the chain rule to differentiate ln(y) with respect to x (y'/y) and use the product rule on the right. Then solve for y'.

 

[LCKurtz]

i recently started my calculus III course but i was stuck with the derivative part when i derive a real number to the power x ( like 2^x) and also when deriving x^x. does anybody have an idea or a link in which these kinds of derivatives can be solved. thanks for ur help

2^x = e^{x ln(2)}

 

[develish16]

thx a lot guys that was so helpful