What is the derivative of #y=x^(5x)#?

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Answer 1 · 2016-05-30T10:53:43

#\frac{d}{dx}(x^{5x})=5x^{5x}(\ln (x)+1)#

#\frac{d}{dx}(x^{5x})#

Applying exponent rule,
#a^b=e^{b\ln (a)}#

#x^{5x}=e^{5x\ln (x)}#

#=\frac{d}{dx}(e^{5x\ln (x)})#

Applying chain rule,
#\frac{df(u)}{dx}=\frac{df}{du}\cdot \frac{du}{dx}#

Let #5x\ln (x)=u#

#=\frac{d}{du}(e^u)\frac{d}{dx}(5x\ln (x))#

We know,
#\frac{d}{du}(e^u)=e^u#
and,
#\frac{d}{dx}(5x\ln (x))=5(\ln (x)+1)#

So,
#\frac{d}{dx}(5x\ln (x))=5(\ln (x)+1)#

Substituting back,
#u=5x\ln (x)#

Simplifying,
#\frac{d}{dx}(x^{5x})=5x^{5x}(\ln (x)+1)#