Final answer:
To find dy/dx, differentiate both sides of the equation. Apply the chain rule and product rule. Isolate (dy/dx) to solve for it.
Step-by-step explanation:
To find dy/dx, we need to differentiate both sides of the given equation with respect to x. Let's start with the left-hand side of the equation:
9 tan⁻¹(x²y)
Now, applying the chain rule, we have:
(d/dx) (9 tan⁻¹(x²y)) = (d/dt)(9 tan⁻¹(t)) * (d/dx)(x²y)
Using the derivative of tan⁻¹(t) and applying the product rule for the derivative of x²y, we get:
(d/dx) (9 tan⁻¹(x²y)) = 9 * (1/(1+(x²y)²)) * (2xy + x²(dy/dx))
Next, let's differentiate the right-hand side of the equation:
(d/dx)(x xy²) = (d/dx)(x) * (xy²) + x * (d/dx)(xy²)
Simplifying this expression, we get:
(d/dx)(x xy²) = y² + 2xy * (dy/dx)
Finally, equating the two derivatives and solving for (dy/dx), we have:
9 * (1/(1+(x²y)²)) * (2xy + x²(dy/dx)) = y² + 2xy * (dy/dx)
Now, we can isolate (dy/dx) by moving all terms involving (dy/dx) to one side:
(9 * (1/(1+(x²y)²)) * x²(dy/dx)) - (2xy * (dy/dx)) = (y² - 9 * (1/(1+(x²y)²)) * 2xy)
Factoring out (dy/dx) from the left-hand side, we get:
((9 * (1/(1+(x²y)²)) * x²) - (2xy)) * (dy/dx) = (y² - 9 * (1/(1+(x²y)²)) * 2xy)
Finally, we can solve for (dy/dx) by dividing both sides by ((9 * (1/(1+(x²y)²)) * x²) - (2xy)):
(dy/dx) = (y² - 9 * (1/(1+(x²y)²)) * 2xy) / ((9 * (1/(1+(x²y)²)) * x²) - (2xy))