Final answer:
To find dy/dx, differentiate the given equation with respect to x. Rearrange the equation to isolate y, then differentiate using the power rule and product rule. Finally, solve for dy/dx by dividing the entire equation by (28y³ + 3yx² + x³).
Step-by-step explanation:
To find dy/dx, we need to differentiate the given equation with respect to x. Let's start by rearranging the equation to isolate y:
7y⁴ + x³y + 6x = 4
Subtracting 6x from both sides:
7y⁴ + x³y = 4 - 6x
Now, we can differentiate both sides with respect to x:
28y³(dy/dx) + 3yx²(dy/dx) + x³(dy/dx) = -6
Factoring out dy/dx:
(28y³ + 3yx² + x³)(dy/dx) = -6
Dividing both sides by (28y³ + 3yx² + x³):
dy/dx = -6 / (28y³ + 3yx² + x³)