If cos(3 x)�6Ó14 x e^2 y=0, find [ dy/dx] using implicit differentiation.

In this problem, you should differentiate both sides of this equation with respect to x. First, using implicit differentiation, differentiate the left side of the equation with respect to x, and express your answer in terms of x,y and [ dy/dx]. To enter your answer to Webwork, every occurence of [ dy/dx] in your answer should be replaced by D, (upper case). For example, if your answer were x(x+y2)[ dy/dx], you would submit x(x+y2)D.

Derivative of left side = .

Now, to find [ dy/dx], set your answer above to 0 and solve for D.

[ dy/dx]=

To find [dy/dx] using implicit differentiation, we first need to differentiate both sides of the equation with respect to x.

Starting with the left side of the equation:
Derivative of cos(3x) is -3sin(3x) by the chain rule.
Derivative of 6y is 6(dy/dx) by the product rule.
Derivative of 14x is 14.
Derivative of e^2y is 2e^2y(dy/dx) by the chain rule.

Putting it all together, the derivative of the left side of the equation is:
-3sin(3x) + 6(dy/dx) + 14 - 2e^2y(dy/dx).

Now, setting our answer equal to 0 to solve for [dy/dx]:
-3sin(3x) + 6(dy/dx) + 14 - 2e^2y(dy/dx) = 0.

To find [dy/dx], isolate the term with [dy/dx] on one side:
6(dy/dx) - 2e^2y(dy/dx) = 3sin(3x) - 14.

Factor out [dy/dx]:
(dy/dx)(6 - 2e^2y) = 3sin(3x) - 14.

Now, solve for [dy/dx]:
[dy/dx] = (3sin(3x) - 14) / (6 - 2e^2y).

Remember to replace [dy/dx] with D (uppercase) when entering your answer into Webwork.