A curve has equation $$\(y=x^{3} \mathrm{e}^{0.2 x}\)$$ where $$\(x \geqslant 0\)$$. At the point $$\(P\)$$ on the curve, the gradient of the curve is 15 . Show that the $$\(x\)$$-coordinate of $$\(P\)$$ satisfies the equation $$\(x=\sqrt{\frac{75 \mathrm{e}^{-0.2 x}}{15+x}}\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_m20_qp_22 Year：2020 Question No：6(a)
Answer:
Differentiate using the product rule
Obtain \(3 x^{2} \mathrm{e}^{0.2 x}+0.2 x^{3} \mathrm{e}^{0.2 x}\)
Equate first derivative to 15 and rearrange to \(x=\ldots\)
Confirm \(x=\sqrt{\frac{75 \mathrm{e}^{-0.2 x}}{15+x}}\)
Obtain \(3 x^{2} \mathrm{e}^{0.2 x}+0.2 x^{3} \mathrm{e}^{0.2 x}\)
Equate first derivative to 15 and rearrange to \(x=\ldots\)
Confirm \(x=\sqrt{\frac{75 \mathrm{e}^{-0.2 x}}{15+x}}\)
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
Solution:
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