Q. If 5=−xy3−3y3−3x then find dxdy in terms of x and y.Answer: dxdy=
Given Equation: We are given the equation 5=−xy3−3y3−3x, and we need to find the derivative of y with respect to x, which is dxdy. To do this, we will use implicit differentiation, treating y as a function of x.
Implicit Differentiation: Differentiate both sides of the equation with respect to x. Remember that when differentiating terms with y, we treat y as a function of x and use the chain rule to include dxdy.dxd[5]=dxd[−xy3]+dxd[−3y3]+dxd[−3x]
Product Rule: The derivative of a constant is 0, so dxd[5]=0. For dxd[−xy3], we use the product rule: dxd[uv]=u′v+uv′, where u=x and v=y3. So, dxd[−xy3]=−1×y3+−x×3y2×dxdy. For dxd[−3y3], we simply multiply the derivative of y3 by −3, which gives us dxd[5]=00. For dxd[5]=01, the derivative is simply −3.
Combining Differentiated Terms: Now we combine all the differentiated terms:0=−y3−3x⋅y2⋅dxdy−9y2⋅dxdy−3.
Isolating Terms: We need to solve for dxdy. First, we isolate the terms that contain dxdy on one side of the equation:3x⋅y2⋅dxdy+9y2⋅dxdy=y3−3.
Factoring Out: Factor out (dxdy) from the left side of the equation:dxdy⋅(3x⋅y2+9y2)=y3−3.
Dividing Both Sides: Now, divide both sides by (3x⋅y2+9y2) to solve for dxdy:dxdy=3x⋅y2+9y2y3−3.
Simplifying the Denominator: We can simplify the denominator by factoring out 3y2:dxdy=3y2⋅(x+3)y3−3.