logarithmic differentiation

Question

Find dy/dx using the method of logarithmic differentiation when y=(3+7x^2)^ln(x)

Answer 1

Take the natural log of both sides to begin: lny = ln[(3 + 7x²)^lnx]

Use log rules to bring lnx in front: lny = lnx·ln(3 + 7x²)

Differentiate implicitly, using product rule: 1/y·dy/dx = ln(3 + 7x²) / x + 14x·lnx / (3 + 7x²)

Multiply both sides by y and replace y with the original expression in x: dy/dx = [ln(3 + 7x²) / x + 14x·lnx / (3 + 7x²)]·[(3 + 7x²)^lnx]