Find the Derivative - d/dx natural log of xy

Problem

d(ln(x*y))/d(x)

Solution

1. Apply the chain rule for the natural logarithm, which states that d(ln(u))/d(x)=1/u⋅d(u)/d(x)

2. Identify the inner function as u=x*y

3. Apply the product rule to differentiate x*y with respect to x treating y as a function of x (implicit differentiation).

4. Calculate the derivative of the inner function:

(d(x)*y)/d(x)=xd(y)/d(x)+yd(x)/d(x)

5. Simplify the derivative of the inner function:

(d(x)*y)/d(x)=xd(y)/d(x)+y

6. Combine the results by multiplying the reciprocal of the inner function by its derivative:

1/(x*y)⋅(xd(y)/d(x)+y)

7. Distribute the fraction to simplify the expression:

(xd(y)/d(x))/(x*y)+y/(x*y)

8. Cancel common terms in the numerator and denominator:

1/yd(y)/d(x)+1/x

Final Answer

d(ln(x*y))/d(x)=1/x+1/yd(y)/d(x)

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