Find the Derivative - d/dx y = natural log of 10/x

Problem

d()/d(x)*ln(10/x)

Solution

1. Apply logarithmic properties to simplify the expression before differentiating. Use the rule ln(a/b)=ln(a)−ln(b)

ln(10/x)=ln(10)−ln(x)

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

d()/d(x)*(ln(10)−ln(x))=d(ln(10))/d(x)−d(ln(x))/d(x)

3. Differentiate the constant term ln(10) Since ln(10) is a constant value, its derivative with respect to x is zero.

d(ln(10))/d(x)=0

4. Differentiate the natural log term using the standard power rule for logarithms, where d(ln(x))/d(x)=1/x

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

5. Combine the results to find the final derivative.

0−1/x=−1/x

Final Answer

d()/d(x)*ln(10/x)=−1/x

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