its difficulty problem.
find dy/dx by implicit differentiation
1.4 cos x sin y = 1
2.tan(x/y)= x+y
1.) 4 cos x sin y = 1
2.) tan(x/y)= x+y
using product rule and chain rule,
4 (-sinx)(siny) + 4(cosx)(cosy y') = 0
y' = sinx*siny / cosx*cosy
= tanx * tany
--------------------
sec^2(x/y)*(1/y - x/y^2 * y') = 1 + y'
y*sec^2(x/y) - y^2
--------------------- = y'
y^2 + x*sec^2(x/y)
To find dy/dx by implicit differentiation, follow these steps:
1. Identify the equation that involves both x and y.
2. Apply the derivative to both sides of the equation with respect to x.
3. When taking the derivative of terms involving y, treat y as a function of x and use the chain rule to differentiate.
4. Solve the resulting equation for dy/dx.
Let's apply these steps to the given problems:
1. 1.4 cos x sin y = 1
Step 1: The equation that involves both x and y is 1.4 cos x sin y = 1.
Step 2: Take the derivative of both sides with respect to x:
d/dx (1.4 cos x sin y) = d/dx (1)
Step 3: Differentiate each term using the product rule and chain rule:
1.4[-sin x sin y dy/dx + cos x cos y] = 0
Step 4: Solve for dy/dx:
-1.4sin x sin y dy/dx + 1.4cos x cos y = 0
dy/dx = (1.4cos x cos y) / (1.4sin x sin y)
dy/dx = cos x / sin x
dy/dx = cot x
Therefore, the derivative dy/dx for the given equation is cot x.
2. tan(x/y) = x + y
Step 1: The equation that involves both x and y is tan(x/y) = x + y.
Step 2: Take the derivative of both sides with respect to x:
d/dx (tan (x/y)) = d/dx (x + y)
Step 3: Differentiate each term using the chain rule:
sec^2 (x/y) * (1/y) * d(x/y)/dx = 1
Step 4: Solve for dy/dx:
sec^2 (x/y) * (1/y) * (1/y) * dy/dx = 1
dy/dx = y^2 / (sec^2 (x/y) * (1/y))
Therefore, the derivative dy/dx for the given equation is y^2 / (sec^2 (x/y) * (1/y)).