國際數學圖譜

IAL 2022 Oct Q4

A Level / Edexcel / P3

IAL 2022 Oct Paper · Question 4

题目

Problem

y=\log_{10}(2x+1)

(a) Express $x$ in terms of $y$ .

(2)

(b) Hence, giving your answer in terms of $x$ , find

\frac{dy}{dx}

(3)

题目中文翻译

y=\log_{10}(2x+1)

(a) 用 $y$ 表示 $x$ 。

(b) 由此用 $x$ 表示并求

\frac{dy}{dx}

解答

(a)

We have

y=\log_{10}(2x+1)

Rewrite this in exponential form:

10^y=2x+1

So

2x=10^y-1

Therefore

\boxed{x=\frac{10^y-1}{2}}

(b)

From part (a),

x=\frac{10^y-1}{2}

Differentiate both sides with respect to $y$ :

\frac{dx}{dy} =\frac12\cdot 10^y\ln10

So

\frac{dx}{dy} =\frac{10^y\ln10}{2}

Therefore

\frac{dy}{dx} =\frac{1}{\frac{10^y\ln10}{2}}

Thus

\frac{dy}{dx} =\frac{2}{10^y\ln10}

Since

10^y=2x+1

we get

\boxed{\frac{dy}{dx}=\frac{2}{(2x+1)\ln10}}