Find the Derivative - d/dx y = natural log of 4x^3-x^2

Problem

d()/d(x)*ln(4*x^3−x^2)

Solution

1. Identify the outer function as ƒ(u)=ln(u) and the inner function as u=4*x^3−x^2

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 u=4*x^3−x^2 using the power rule to get d(u)/d(x)=12*x^2−2*x

4. Substitute these components into the chain rule formula.

d(ln(4*x^3−x^2))/d(x)=1/(4*x^3−x^2)⋅(12*x^2−2*x)

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

d(ln(4*x^3−x^2))/d(x)=(2*x*(6*x−1))/(x^2*(4*x−1))

6. Cancel the common factor of x

d(ln(4*x^3−x^2))/d(x)=(2*(6*x−1))/(x*(4*x−1))

Final Answer

d(ln(4*x^3−x^2))/d(x)=(12*x−2)/(4*x^3−x^2)

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