Find the Derivative - d/dx x/( natural log of x)

Problem

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

Solution

1. Identify the rule needed for differentiation. Since the expression is a fraction of two functions, apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the variables for the numerator and denominator. Let u=x and v=ln(x)

3. Differentiate the individual components. The derivative of the numerator is d(x)/d(x)=1 and the derivative of the denominator is d(ln(x))/d(x)=1/x

4. Substitute these values into the quotient rule formula.

d()/d(x)x/ln(x)=(ln(x)⋅1−x⋅1/x)/((ln(x))2)

5. Simplify the numerator by performing the multiplication x⋅1/x=1

d()/d(x)x/ln(x)=(ln(x)−1)/((ln(x))2)

Final Answer

d()/d(x)x/ln(x)=(ln(x)−1)/((ln(x))2)

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