A curve has parametric equations $$\[ x=\ln (2 t+6)-\ln t, \quad y=t \ln t . \]$$ Find the exact gradient of the curve at $$$P$$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_s21_qp_21 Year：2021 Question No：4(b)

Answer:

Obtain $\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{2}{2 t+6}-\frac{1}{t}$
Use product rule to find $\frac{\mathrm{d} y}{\mathrm{~d} t}$
Obtain $\ln t+t \times \frac{1}{t}$
Divide to obtain $\frac{\mathrm{d} y}{\mathrm{~d} x}$ using their $\frac{\mathrm{d} y}{\mathrm{~d} t}$ and $\frac{\mathrm{d} x}{\mathrm{~d} t}$ correctly
Obtain $-6(\ln 3+1)$

Knowledge points:

2.4.1 use the derivatives of ,ln x,sin x,cos x,tan x, together with constant multiples, sums, differences and composites

2.4.2 differentiate products and quotients

2.4.3 find and use the first derivative of a function which is defined parametrically or implicitly.

Answer:

Knowledge points:

Solution:

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