Find dy/dx cos(xy)+x^7=y^7

Problem

cos(x*y)+x7=y7

Solution

1. Differentiate both sides with respect to x treating y as a function of x

d(cos(x*y))/d(x)+d(x7)/d(x)=d(y7)/d(x)

2. Apply the chain rule and the product rule to the first term, and the power rule to the others.

−sin(x*y)⋅(d(x)*y)/d(x)+7*x6=7*y6d(y)/d(x)

3. Apply the product rule to the term (d(x)*y)/d(x)

−sin(x*y)*(y+xd(y)/d(x))+7*x6=7*y6d(y)/d(x)

4. Distribute the −sin(x*y) term.

−y*sin(x*y)−x*sin(x*y)d(y)/d(x)+7*x6=7*y6d(y)/d(x)

5. Group all terms containing d(y)/d(x) on one side of the equation.

7*x6−y*sin(x*y)=7*y6d(y)/d(x)+x*sin(x*y)d(y)/d(x)

6. Factor out d(y)/d(x) from the right side.

7*x6−y*sin(x*y)=d(y)/d(x)*(7*y6+x*sin(x*y))

7. Solve for dy/dx by dividing both sides by the expression in parentheses.

d(y)/d(x)=(7*x6−y*sin(x*y))/(7*y6+x*sin(x*y))

Final Answer

d(y)/d(x)=(7*x6−y*sin(x*y))/(7*y6+x*sin(x*y))

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