A curve has equation $$\[ 3 x^{2}-y^{2}-4 \ln (2 y+3)=26 \text {. } \]$$ Find the equation of the tangent to the curve at the point $$\((3,-1)\)$$. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................
Exam No：9709_m20_qp_22 Year：2020 Question No：4
Answer:
Differentiate \(-y^{2}\) to obtain \(-2 y \frac{\mathrm{d} y}{\mathrm{~d} x}\)
Differentiate \(-4 \ln (2 y+3)\) to obtain \(\frac{-8}{2 y+3} \frac{\mathrm{d} y}{\mathrm{~d} x}\)
Attempt differentiation of all terms
Substitute \(x=3, y=-1\) to find numerical value of \(\frac{\mathrm{d} y}{\mathrm{~d} x}\)
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=3\)
Obtain equation \(y=3 x-10\)
Differentiate \(-4 \ln (2 y+3)\) to obtain \(\frac{-8}{2 y+3} \frac{\mathrm{d} y}{\mathrm{~d} x}\)
Attempt differentiation of all terms
Substitute \(x=3, y=-1\) to find numerical value of \(\frac{\mathrm{d} y}{\mathrm{~d} x}\)
Obtain \(\frac{\mathrm{d} y}{\mathrm{~d} x}=3\)
Obtain equation \(y=3 x-10\)
Knowledge points:
1.7.3 apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change (Including connected rates of change, e.g. given the rate of increase of the radius of a circle, find the rate of increase of the area for a specific value of one of the variables.)
2.4.1 use the derivatives of ,ln x,sin x,cos x,tan x, together with constant multiples, sums, differences and composites
2.4.3 find and use the first derivative of a function which is defined parametrically or implicitly.
Solution:
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