题目

Problem

In this question you must show all stages of your working.

(a) Write

\begin{align*} y=\frac{5x^2+\sqrt{x^3}}{\sqrt[3]{8x}} \end{align*}

in the form

\begin{align*} y=Ax^p+Bx^q \end{align*}

where $A$ , $B$ , $p$ and $q$ are constants to be found.

(4)

(b) Hence find $\dfrac{dy}{dx}$ giving each coefficient in simplest form.

(3)

解答

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把根式全部改写成指数，再分别除以分母。这里 $\sqrt[3]{8x}=2x^{1/3}$ 。

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\begin{align*} y =&\,\frac{5x^2+\sqrt{x^3}}{\sqrt[3]{8x}}\\ =&\,\frac{5x^2+x^{\frac32}}{2x^{\frac13}}\\ =&\,\frac{5x^2}{2x^{\frac13}} +\frac{x^{\frac32}}{2x^{\frac13}}\\ =&\,\frac52x^{2-\frac13} +\frac12x^{\frac32-\frac13}\\ =&\,\frac52x^{\frac53}+\frac12x^{\frac76}. \end{align*}

Therefore

\begin{align*} A=\frac52,\qquad p=\frac53,\qquad B=\frac12,\qquad q=\frac76. \end{align*}

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直接对 (a) 的幂函数形式逐项求导。

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From part (a),

\begin{align*} y=\frac52x^{\frac53}+\frac12x^{\frac76}. \end{align*}

Differentiate:

\begin{align*} \frac{dy}{dx} =&\,\frac52\cdot\frac53x^{\frac23} +\frac12\cdot\frac76x^{\frac16}\\ =&\,\frac{25}{6}x^{\frac23}+\frac{7}{12}x^{\frac16}. \end{align*}