Find the Derivative - d/dx natural log of 8x

Problem

d(ln(8*x))/d(x)

Solution

1. Identify the function as a composition of the natural logarithm ln(u) and the linear function u=8*x

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

3. Differentiate the inner function u=8*x with respect to x to get (d(8)*x)/d(x)=8

4. Substitute the components into the chain rule formula: 1/(8*x)⋅8

5. Simplify the expression by canceling the constant factor 8 from the numerator and denominator.

Final Answer

d(ln(8*x))/d(x)=1/x

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