Use logarithmic differentiation to find dy/dx: y=x^1/x

Question

Use logarithmic differentiation

Answer 1

y= x^1/x

lny = ln x^1/x

lny = 1/x lnx = lnx/x

Now take the derivative of both sides with respect to x.

1/y y' = (x 1/x - lnx (1))/x² (use the quotient rule on the right side)

y' = (1 - lnx)/x² (y) = (1-lnx)/x² (x^1/x) (at this point you can use Law of Exponents to write x^{(1/x -2) in the numerator)}

Check out this graph for confirmation. Note that when you enable the functions g and h, they both have the same graph. Use the slider on x₁ to see different flavors of the tangent line to f(x).

desmos.com/calculator/jm2ttqidc0