calculus question

Question

1- Given the question below find dy/dx.

38x⁶+10x⁵⁷y+y⁹=49

dy/dx= -228x⁵+570x⁵⁶y/10x⁵⁷+9y⁸

2- Now, find the equation of the tangent line to the curve at (1, 1). Write your answer in mx+b format

y=?

Answer 1

Hi there!

The goal here is to write the equation of the tangent line to the curve at the point (x₁,y₁)=(1,1).

Let's consider the point-slope form of a line: y-y₁=m(x-x₁), where (x₁,y₁)=(1,1) is a point on the line. Then we have: y-1=m(x-1). We then need to find m, the slope. Luckily we already have the derivative:

m = dy/dx = -(228x⁵ + 570x⁵⁶y)/(10x⁵⁷+9y⁸)

To obtain the value at (x₁,y₁)=(1,1), we "plug in" to find

m = -(228 + 570)/(10+9) = -42

Thus, our equation is: y-1=-42(x-1), which simplifies to y=-42x+43, our final answer.

Please feel free to reach out if you have any more questions! Good luck!