use implicit differentiation to find the slope of the tangent line to the curve

3^x + log_2(xy) = 10

at the point (2,1) and use it to find the equation of the tangent line in the form y = mx + b

Question

use implicit differentiation to find the slope of the tangent line to the curve

3^x + log_2(xy) = 10

at the point (2,1) and use it to find the equation of the tangent line in the form y = mx + b

experimentx · Answer

differentiate it and get the differential equation 3^x*ln(3) + 1/(xy)*{1+xdy/dx} = 0 find value of dy/dx which is .. roughly -21 http://www.wolframalpha.com/input/?i=3%5E2*ln%283%29+%2B+1%2F2%281%2Bx%29+%3D+0 which is the slope of the tangent, since you know slope m, and you have point (2,1) find the value of b, you have your tangent

experimentx · Answer

oops sorry, slope is around -1.05 http://www.wolframalpha.com/input/?i=3%5E2*ln%283%29+%2B+1%2F%282%281%2Bx%29%29+%3D+0

experimentx · Answer

is that log base 2??

anonymous · Answer

yes it is.

anonymous · Answer

I got this: $$(\ln 3)3^x+\frac{1}{(\ln 2)xy}\left(x\frac{dy}{dx}+y\right)=0$$

anonymous · Answer

you can solve for $$\frac{dy}{dx}$$ and substitute the coordinates given. so you'll have the slope.