Find an equation of the line tangent to the curve defined by x^5+6xy+y^4=72 at the point(2,2).
+129845 x^5+6xy+y^4=72 at the point(2,2)
Using implicit differenttiation, we have
5x^4 + 6y + 6xy' + 4y^3y' = 0
y ' [ 6x + 4y^3 ] = - [ 5x^4 + 6y]
y ' = - [ 5x^4 + 6y] / [ 6x + 4y^3]
The slope at [2,2] = - [ 5(2)^4 + 6(2) ] / [[ 6(2) + 4(2)^3] = - [ 80 + 12] / [ 12 + 32] = [- [92 / 44] =
-[23/11]
So the equation of the tangent line is
y = -(23/11)(x - 2) + 2 =
y = -(23/11)x + 46/11 + 2
y = -(23/11)x + 68/11
Here's the graph : https://www.desmos.com/calculator/man5p1joko
