Final answer:
To find dy/dx of the given equation, implicit differentiation is used and results in dy/dx = (2x + 5y sin(x)) / (5cos(x) - 2y).
Step-by-step explanation:
To find dy/dx of the equation 5y cos(x) = x² + y², we need to differentiate both sides of the equation with respect to x using implicit differentiation. This involves taking the derivative of each term separately and applying the product rule to the term involving both x and y.
First, we differentiate 5y cos(x) with respect to x:
d/dx[5y cos(x)] = 5y * d/dx[cos(x)] + cos(x) * d/dx[5y] = -5y sin(x) + 5cos(x) dy/dx
Next, we differentiate x²+ y² with respect to x:
d/dx[x² + y²] = 2x + 2y dy/dx
Now, we equate these two expressions:
-5y sin(x) + 5cos(x) dy/dx = 2x + 2y dy/dx
To isolate dy/dx, we combine the terms involving dy/dx on one side and the rest on the other side:
5cos(x) dy/dx - 2y dy/dx = 2x + 5y sin(x)
dy/dx(5cos(x) - 2y) = 2x + 5y sin(x)
Finally, we solve for dy/dx:
dy/dx = (2x + 5y sin(x)) / (5cos(x) - 2y)