Determine the slope of the graph of x4 = ln(xy) at the point (1, e).

Answer-

The slope of the graph is 3e

Solution-

The given equation-

Using logarithm properties,

Taking derivatives of both sides,

Applying chain rule,

Slope of at is

step-by-step explanation:

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4