What is the implicit differentiation of x+y^2=cos(xy)?
To find the implicit differentiation of the given equation x+y^2=cos(xy), we differentiate both sides of the equation with respect to x using the rules of differentiation. Here's how you can do it step by step:
Step 1: Differentiate both sides of the equation with respect to x, treating y as a function of x.
d/dx(x) + d/dx(y^2) = d/dx(cos(xy))
Step 2: Differentiate each term on the left side separately.
1 + 2y(dy/dx) = d/dx(cos(xy))
Step 3: Now, differentiate the right side using the chain rule.
1 + 2y(dy/dx) = -sin(xy)(d/dx(xy))
Step 4: Apply the rules of differentiation to the derivative of the product xy.
1 + 2y(dy/dx) = -sin(xy)(y + x(dy/dx))
Step 5: Simplify the equation by distributing the -sin(xy) term.
1 + 2y(dy/dx) = -y*sin(xy) - x(dy/dx)*sin(xy)
Step 6: Gather the terms involving dy/dx on one side.
2y(dy/dx) + x(dy/dx)*sin(xy) = -y*sin(xy) - 1
Step 7: Factor out dy/dx on the left side.
dy/dx(2y + x*sin(xy)) = -y*sin(xy) - 1
Step 8: Divide both sides by (2y + x*sin(xy)) to solve for dy/dx.
dy/dx = (-y*sin(xy) - 1) / (2y + x*sin(xy))
That is the implicit differentiation of the equation x+y^2=cos(xy).