The accounting equation is: Assets = Capital + Liabilities. When Mahnoor brings her car into the business, the business gains an asset (the car), and simultaneously the owner's capital increases by the same amount (as it is a capital contribution). No liability is created and no profit is involved. So both Assets and Capital increase by the same value, keeping the equation balanced.

Write $\frac{6x\sqrt{x}}{1-9x^3} = \frac{2\cdot3x^{3/2}}{1-(3x^{3/2})^2}=\frac{2t}{1-t^2}$ where $t=3x^{3/2}$.

So $\tan^{-1}\!\left(\frac{2t}{1-t^2}\right)=2\tan^{-1}(t)=2\tan^{-1}(3x^{3/2})$

Differentiate: $\frac{d}{dx}[2\tan^{-1}(3x^{3/2})] = \frac{2}{1+9x^3}\cdot 3\cdot\frac{3}{2}x^{1/2} = \frac{9\sqrt{x}}{1+9x^3}$

Since this equals $\sqrt{x}\cdot g(x)$: $g(x)=\dfrac{9}{1+9x^3}$

Uterine Rupture — a catastrophic obstetric emergency. Classic signs (mnemonic: ABCDE of rupture):

• Abdominal pain — sudden, severe, tearing/ripping (contractions may cease)
• Bleeding (intraperitoneal — minimal external bleeding)
• Cessing of contractions (previously regular contractions stop)
• Disappearing presenting part (head recedes from pelvis into abdomen)
• Emergency fetal distress (sudden prolonged bradycardia)

Additional: Maternal shock (tachycardia, hypotension), loss of uterine contour, tender abdomen.

Risk factors:

• Previous uterine scar (LSCS, myomectomy)
• Grand multiparity
• Prolonged obstructed labor
• Oxytocin misuse
• Instrumental delivery

Management: Call crash team → IV access × 2, fluids, crossmatch → Immediate laparotomy → Primary repair or hysterectomy depending on extent → Deliver baby (perinatal mortality 50–75% in complete rupture).

Incidence of rupture in scarred uterus: \(\approx 0.5{-}1\%\) with LSCS scar (higher with classical CS scar \(\approx 4{-}9\%\)).