What is the derivative of x^(10x)?

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Answer 1 · 2018-01-25T17:01:12

#dy/dx=(10ln(x)+10)x^(10x)#

#y=x^(10x)#

Take the logarithm of both sides then use implicit differentiation:

#lny=ln(x^(10x))#

#->lny=10xln(x)# (Using the laws of logs)

Now implicitly differentiate via the product rule:

#1/ydy/dx=10ln(x)+10x*1/x=10ln(x)+10#

Multiply by #y#:

#dy/dx=(10ln(x)+10)y#

Substitute #y# back in:

#dy/dx=(10ln(x)+10)x^(10x)#