Differentiate each of the following with respect to $x$: $\ln\left(\dfrac{12x-7}{2x^2+5}\right)$

Question

Differentiate each of the following with respect to $x$: $\ln\left(\dfrac{12x-7}{2x^2+5}\right)$[2] $\tan^{-1}(xe^x)$[3] The curve $C$ has equation $2x-y^2=x^2y$. Find the equation of the tangent to $C$ which is parallel to the $x$-axis.[4] The line $l$ is normal to $C$ at $(1,-2)$. Without using a calculator, determine if line $l$ intersects curve $C$ again.[4]

Accepted Answer

(a)(i) $\dfrac{4(-6x^2+7x+15)}{(12x-7)(2x^2+5)}$ (ii) $\dfrac{e^x(1+x)}{1+x^2e^{2x}}$. (b)(i) $y=1$. (ii) Yes, at two further points; $l:y=\dfrac x2-\dfrac52$.

2026 CJC Promo Q9

2026 CJC Promo Q9

Solution:

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