Q. Using implicit differentiation, find dxdy.xy=7+x2y2
Differentiate with respect to x: Differentiate both sides of the equation with respect to x using implicit differentiation.The left side of the equation is xy, which is (xy)1/2. Using the chain rule, the derivative of (xy)1/2 with respect to x is (1/2)(xy)−1/2∗(y+xdxdy).The right side of the equation is 7+x2∗y2. The derivative of 7 with respect to x is x0. The derivative of x1 with respect to x is x3 using the product rule.So, differentiating both sides gives us:x4.
Simplify the equation: Simplify the differentiated equation.(21)(xy)−21∗(y+xdxdy)=2x∗y2+2x2∗ydxdy.Multiply both sides by 2(xy)21 to get rid of the fraction and the square root:y+xdxdy=4x∗y2∗(xy)21+4x2∗y∗(xy)21∗dxdy.
Isolate terms: Isolate terms with dxdy on one side and the rest on the other side.xdxdy−4x2y(xy)21dxdy=4xy2(xy)21−y.
Factor out dxdy: Factor out dxdy on the left side of the equation.dxdy(x−4x2⋅y⋅(xy)21)=4x⋅y2⋅(xy)21−y.
Solve for (dxdy): Solve for (dxdy).dxdy=x−4x2⋅y⋅(xy)214x⋅y2⋅(xy)21−y.
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