Find the Derivative - d/dx natural log of x/7

Problem

d(ln(x/7))/d(x)

Solution

1. Apply the property of logarithms to simplify the expression before differentiating. Use the rule ln(a/b)=ln(a)−ln(b)

ln(x/7)=ln(x)−ln(7)

2. Apply the sum rule for derivatives, which states that the derivative of a sum/difference is the sum/difference of the derivatives.

d(ln(x)−ln(7))/d(x)=d(ln(x))/d(x)−d(ln(7))/d(x)

3. Differentiate the first term using the standard power rule for natural logarithms, where the derivative of ln(x) is 1/x

d(ln(x))/d(x)=1/x

4. Differentiate the second term noting that ln(7) is a constant. The derivative of any constant is 0

d(ln(7))/d(x)=0

5. Combine the results to find the final derivative.

1/x−0=1/x

Final Answer

d(ln(x/7))/d(x)=1/x

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