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

Problem

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

Solution

1. Identify the function as a composition of the natural logarithm and a linear function, requiring the use of the chain rule.

2. Apply the chain rule which states that d(ln(u))/d(x)=1/u⋅d(u)/d(x) where u=8*x

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

4. Substitute these values into the chain rule formula.

d(ln(8*x))/d(x)=1/(8*x)⋅8

5. Simplify the expression by canceling the common factor of 8 in the numerator and denominator.

8/(8*x)=1/x

Final Answer

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

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