Find the Derivative - d/dx y = natural log of 9x^3-x^2

Problem

d()/d(x)*ln(9*x3−x2)

Solution

1. Identify the outer and inner functions for the chain rule. The outer function is ln(u) and the inner function is u=9*x3−x2

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

3. Differentiate the inner function 9*x3−x2 with respect to x using the power rule.

d(9*x3−x2)/d(x)=27*x2−2*x

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

d(y)/d(x)=1/(9*x3−x2)⋅(27*x2−2*x)

5. Simplify the expression by multiplying the terms and factoring out x from the numerator and denominator.

d(y)/d(x)=(x*(27*x−2))/(x*(9*x2−x))

6. Cancel the common factor of x to reach the final simplified form.

d(y)/d(x)=(27*x−2)/(9*x2−x)

Final Answer

d(ln(9*x3−x2))/d(x)=(27*x−2)/(9*x2−x)

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