Question

For each problem use implicit differentiation to find DY/DX in terms of X and Y

2) 3x^2+3=sec2y^3


A) dy/dx= 3x^2+3/sec2y^3

B) dy/dx= y^2sec2y^3tan2y3/x

C) dy/dx= -6x/sec2y^3

D) dy/dx= x/y^2sec2y^3tan2y3​

Answer 1

To find the derivative dy/dx by implicit differentiation for the equation 3x^2 + 3 = sec(2y^3), one must apply the chain rule and then rearrange the terms. The correct answer is dy/dx = -6x/(sec(2y^3)tan(2y^3)−3y^2).

Step-by-step explanation:

To find dy/dx using implicit differentiation, we start with the given equation, 3x^2 + 3 = sec(2y^3). First, differentiate both sides of the equation concerning x. The left-hand side, 3x^2 + 3, differentiates to 6x. The right-hand side, sec(2y^3), is a composite function and requires the chain rule. The differentiation gives us sec(2y^3)tan(2y^3)−3y^2, where represents dy/dx. After differentiating, we rearrange the equation to solve for dy/dx.

Using these steps, we find that the correct answer is dy/dx = -6x/(sec(2y^3)tan(2y^3)−3y^2), which corresponds to option C: dy/dx = -6x/sec(2y^3).