Find dy/dx of 3 sin x cos y = 2 using implicit differentiation.
3 sinx cosy = 2
3 cosx cosy - 3sinx siny y' = 0
y' = cosx cosy / sinx siny = cotx coty
To find dy/dx using implicit differentiation, differentiate both sides of the equation with respect to x, treating y as a function of x.
Differentiating the left side:
We'll need to use the product rule for differentiation.
For the term 3 sin x, the derivative is 3 cos x.
For the term cos y, the derivative is -sin y multiplied by dy/dx.
So, the left side becomes: 3 cos x cos y - 3 sin x sin y * dy/dx.
Differentiating the right side:
The derivative of 2 with respect to x is 0, since it is a constant.
Setting up the equation with the differentiated left and right side:
3 cos x cos y - 3 sin x sin y * dy/dx = 0
Now, solve for dy/dx by isolating the variable:
3 sin x sin y * dy/dx = 3 cos x cos y
Dividing both sides by 3 sin x sin y:
dy/dx = 3 cos x cos y / (3 sin x sin y)
Simplifying the expression:
dy/dx = cos x / sin x
So, the derivative of the equation 3 sin x cos y = 2 with respect to x is dy/dx = cos x / sin x.
To find the derivative of y with respect to x, dy/dx, using implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x.
Differentiating both sides of the equation 3sin(x)cos(y) = 2:
d/dx [3sin(x)cos(y)] = d/dx [2]
Step 2: Apply the chain rule to the left-hand side of the equation.
Let's differentiate the function 3sin(x)cos(y) with respect to x using the chain rule:
d/dx [3sin(x)cos(y)] = (d/dx [3sin(x)])cos(y) + 3sin(x)(d/dx [cos(y)])
Step 3: Evaluate the derivatives of each term.
Differentiating sin(x) with respect to x, we get:
d/dx [sin(x)] = cos(x)
Differentiating cos(y) with respect to y, we get:
d/dy [cos(y)] = -sin(y)
Step 4: Substitute the derivatives back into the equation.
Now, let's substitute the derivatives into the equation:
(cos(x))(cos(y)) + 3sin(x)(-sin(y)) = 0
Step 5: Solve for dy/dx.
To solve for dy/dx, rearrange the terms:
(cos(x))(cos(y)) - 3sin(x)(sin(y)) = 0
Now, isolate (-sin(x))(sin(y)) on one side:
(-sin(x))(sin(y)) = (cos(x))(cos(y))
Finally, divide both sides by (-sin(x))(cos(y)) to get dy/dx:
dy/dx = (cos(x))(cos(y)) / (-sin(x))(sin(y))
Therefore, the derivative dy/dx of the equation 3sin(x)cos(y) = 2 using implicit differentiation is:
dy/dx = (cos(x))(cos(y)) / (-sin(x))(sin(y))