Tangents and Intersection

Question

 Tangents and Intersection. Find the two tangent slopes (exact values) at the point where the graph of the equation y^x = x^y intersects the graph of y = log base 2 (x) Include your computations of both derivatives. See Desmos.

Answer 1

From the graph, the intersection is (4,2).

First derivative:

y^x=x^y

xlny = ylnx

x/y y' + lny =y/x+y'lnx

y'(x/y-lnx) = y/x-lny

y'=(y/x-lny)/(x/y-lnx)

y'(4,2) = (2/4-ln2)/(4/2-ln4)

= (1/2-ln2)/(2-ln4)

Second derivative

y = log₂(x) = lnx/ln2

y' = 1/(xln2)

y'(4) = 1/(4ln2) = 1/ln(2⁴)=1/ln(16)

Answer 2

desmos.com/calculator/zanw4ur2ix