Question Detail

Given y = log_7(4x²), we need to find y' and y''.Assuming log_7 implies ln in the context of the options, or that ln(7) cancels out or is omitted in the options, we can rewrite y using logarithm properties:y = log_7(4x²) = (ln(4x²)) / ln(7) = (ln(4) + ln(x²)) / ln(7) = (ln(4) + 2ln(x)) / ln(7).

1. Find the first derivative y': y' = d/dx [ (ln(4) + 2ln(x)) / ln(7) ] y' = (1/ln(7)) * d/dx [ ln(4) + 2ln(x) ] y' = (1/ln(7)) * (0 + 2/x) = 2 / (x ln(7))
2. Find the second derivative y'': y'' = d/dx [ 2 / (x ln(7)) ] = (2/ln(7)) * d/dx (x⁻¹) y'' = (2/ln(7)) * (-1x⁻²) = -2 / (x² ln(7))
3. Calculate y'' - 3y': y'' - 3y' = -2 / (x² ln(7)) - 3 * (2 / (x ln(7))) = -2 / (x² ln(7)) - 6 / (x ln(7)) To combine these, find a common denominator x² ln(7): = -2 / (x² ln(7)) - (6x) / (x² ln(7)) = (-2 - 6x) / (x² ln(7)) = -(2 + 6x) / (x² ln(7))

Comparing with the options, if the ln(7) factor in the denominator is omitted (possibly a simplification or error in problem statement/options for log base e), then the expression matches option B: -(2+6x)/x².