Find the equation of the tangent line to the graph

Question

graph of

4y² + 7x³ = 5x⁴ - y³ +7y -26 at (3,5)

Answer 1

Hi again,

For this kind of problem where we don't have a "clean" function (y is not alone on one side and we have a bunch of exponents that make dealing with that ugly), we can use implicit differentiation. Implicit differentiation relies on the chain rule for derivatives. Remember we are taking dy/dx in order to come up with a slope of the tangent line when we get to finishing this problem.

First, differentiate both sides of the function with respect to x, and obey the chain rule. Here, everywhere you see a dy/dx in the second line is a consequence of the chain rule.

d/dx (4y² + 7x³) = d/dx (5x⁴ - y³ + 7y - 26)

8y(dy/dx) + 21x² = 20x³ - 3y²(dy/dx) + 7(dy/dx)

Bring all the dy/dx terms to one side of the equation

8y(dy/dx) + 3y²(dy/dx) - 7(dy/dx) = 20x³ - 21x²

factor out the dy/dx

dy/dx (3y² + 8y - 7) = 20x³ - 21x²

dy/dx = (20x³ - 21x²) / (3y² + 8y - 7)

Ok, now just evaluate dy/dx at the point of interest (3,5). This is done by plugging in x=3 and y=5 into the above equation. You should get dy/dx = 3.25 (this is the value of the slope at this point on the curve)

Now, to write the equation of the tangent line, we can use the point-slope equation of a line (x₁ is 3, y₁ is 5)

y - y₁ = m (x - x₁)

y - 5 = 3.25 (x - 3)

Rearrange and multiply it out

y = 3.25x - 4.75

I apologize if I made any typos in transcribing my steps, but this solution here works and makes sense as a tangent line if you graph the function and the line.