How do you implicitly differentiate 6=ylny-x6=ylnyx?

2 Answers
Dec 22, 2016

(dy)/(dx)=1/(lny+1)dydx=1lny+1

Explanation:

differentiate both side wrt xx

d/(dx)(6)=d/(dx)(ylny)-d/(dx)(x)ddx(6)=ddx(ylny)ddx(x)

points to note:

1) When differentiating f(y)f(y) wrt ""x"" x differentiate wrt yy and then multiply by (dy)/(dx)dydx

2) for the first term on RHS the product rule needs to be employed.

0=(dy)/(dx)lny+cancel(yxx1/y)xx(dy)/(dx)-1

0=(dy)/(dx)lny+(dy)/(dx)-1

Now rearrange for (dy)/(dx)

0=(lny+1)(dy)/(dx)-1

(dy)/(dx)=1/(lny+1)

Dec 22, 2016

dy/dx=1/(1+lny)

Explanation:

differentiate color(blue)"implicitly with respect to x"

To differentiate ylny" use " color(blue)"product rule"

#0=(yxx1/y.dy/dx+lnyxx1.dy/dx)-1#

rArrdy/dx(1+lny)=1

rArrdy/dx=1/(1+lny)