Find the Derivative - d/dx y = natural log of 8-9x

Problem

d()/d(x)*ln(8−9*x)

Solution

1. Identify the outer function as the natural logarithm ln(u) and the inner function as u=8−9*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=8−9*x with respect to x

d(8−9*x)/d(x)=−9

4. Substitute the inner function and its derivative back into the chain rule formula.

d(ln(8−9*x))/d(x)=1/(8−9*x)⋅(−9)

5. Simplify the expression by multiplying the terms.

(−9)/(8−9*x)

Final Answer

d(ln(8−9*x))/d(x)=−9/(8−9*x)

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