0606 P13 - Nov 2023 - Q9 - 5 marks
7744
Given that
\(\displaystyle y=\frac{(5x+2)^{\frac13}}{(x-1)^2},\)
show that \(\frac{\mathrm dy}{\mathrm dx}\) can be written in the form
\(\displaystyle -\frac{Ax+B}{3(5x+2)^{\frac23}(x-1)^3},\)
where \(A\) and \(B\) are integers.
Solution
Answer: \(A=25,\ B=17\).
Differentiate first. Stationary points occur where the derivative is zero, while tangent and normal problems use the gradient at the given point.
Write
\(y=(5x+2)^{\frac13}(x-1)^{-2}.\)
Using the product rule,
\(\displaystyle \frac{\mathrm dy}{\mathrm dx}=\frac53(5x+2)^{-\frac23}(x-1)^{-2}-2(5x+2)^{\frac13}(x-1)^{-3}.\)
Use the common denominator
\(3(5x+2)^{\frac23}(x-1)^3.\)
The first term contributes
\(5(x-1)\)
to the numerator.
The second term contributes
\(-6(5x+2)\)
to the numerator.
Therefore the numerator is
\(5(x-1)-6(5x+2)=5x-5-30x-12=-(25x+17).\)
Hence
\(\displaystyle \frac{\mathrm dy}{\mathrm dx}=-\frac{25x+17}{3(5x+2)^{\frac23}(x-1)^3}.\)
So
\(A=25,\quad B=17.\)
