Find dy/dx 4xy+ natural log of x^2y=7

Problem

4*x*y+ln(x2*y)=7

Solution

1. Expand the natural logarithm using the properties of logarithms to make differentiation easier.

ln(x2*y)=ln(x2)+ln(y)

4*x*y+2*ln(x)+ln(y)=7

2. Differentiate both sides of the equation with respect to x using implicit differentiation.

d()/d(x)*(4*x*y+2*ln(x)+ln(y))=d()/d(x)*7

3. Apply the product rule to 4*x*y and the chain rule to ln(y)

4*y+4*xd(y)/d(x)+2/x+1/yd(y)/d(x)=0

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

4*xd(y)/d(x)+1/yd(y)/d(x)=−4*y−2/x

5. Factor out d(y)/d(x) from the left side.

d(y)/d(x)*(4*x+1/y)=−4*y−2/x

6. Solve for d(y)/d(x) by dividing both sides by the expression in the parentheses.

d(y)/d(x)=(−4*y−2/x)/(4*x+1/y)

7. Simplify the complex fraction by multiplying the numerator and denominator by x*y

d(y)/d(x)=(−4*x*y2−2*y)/(4*x2*y+x)

Final Answer

d(y)/d(x)=−(4*x*y2+2*y)/(4*x2*y+x)

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