Find an equation of the tangent line

Question

find an equation of the tangent line of y=x^7x when x=e

Answer 1

y = x^7x

 

Use logarithmic differentiation to find y':

 

lny = ln(x^7x) = 7x(lnx)

 

y'/y = 7lnx + 7x(1/x) = 7lnx + 7

 

y' = y(7lnx + 7) = x^7x(7lnx + 7)

 

When x = e, y' = e^7e(7lne + 7)

 

                      = 14e^7e slope of tangent line

 

Point on tangent line = (e, e^7e)

 

Equation of tangent line:  y - e^7e = 14e^7e(x - e)