Final answer:
To find y by implicit differentiation for the equation 4x² + 3y² = 6, differentiate both sides with respect to x. Use the power rule for differentiation and the chain rule for the y term. Isolate dy/dx to obtain the final result.
Step-by-step explanation:
To find y by implicit differentiation for the equation 4x² + 3y² = 6, we need to differentiate both sides of the equation with respect to x.
- For the left side of the equation, we apply the power rule for differentiation, which states that if we have a term of the form ax^n, its derivative is given by d/dx(ax^n) = nax^(n-1). So, the derivative of 4x² would be 8x.
- For the right side of the equation, we differentiate the constant term, which is 6, resulting in 0.
- For the term 3y², we apply the chain rule. The derivative of y² with respect to x would be 2y * (dy/dx) by the chain rule.
So, applying the above steps, we get:
8x + 2y * (dy/dx) = 0
To find dy/dx, we isolate it by moving the 8x term to the other side:
dy/dx = -8x / (2y)
Simplifying further, we get:
dy/dx = -4x / y