Find the Derivative - d/dx natural log of x^2+9

Problem

d(ln(x2+9))/d(x)

Solution

1. Identify the outer function as ln(u) and the inner function as u=x2+9

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

3. Differentiate the inner function x2+9 with respect to x to get 2*x

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

d(ln(x2+9))/d(x)=1/(x2+9)⋅2*x

5. Simplify the expression by multiplying the terms.

Final Answer

d(ln(x2+9))/d(x)=(2*x)/(x2+9)

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