Find the slope, implicit differentiation (no calculator)

Question

The slope of the tangent line to the graph of ln(x²y) = 2-x² at the point where x = 1 is:

A. -4e

B. -3e

C. 0

D. 2e

Answer 1

Use Log rules to rewrite the natural log

ln(x²y) = 2-x²

2lnx + lny = 2 - x²

(isolate y for later not necessary)

lny = 2 - x² - 2lnx

y = e^(2 - x² - 2lnx)

Implicitly differentiate.

2/x + y'/y = 0 - 2x

y'/y = -2x - 2/x

y' = y(-2x-2/x)

Solve @ x=1

y'= e^(2-1-2ln1) * (-2-2)

y' = e*-4

y' = -4e