Question: The perimeter of a rectangle is 60 cm. Its length is 12 cm. What is its width?
1. Write an expression using x to represent the rectangle's width.
2. Solve for the width.

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Ответ:

kat9323

18.06.2022

Mathematics

$\dfrac{1-3ln x}{x^{4}}$

Step-by-step explanation:

According to Quotient rule in derivatives :

If f(x) and g(x) are differentiable functions , then

$\dfrac{d}{dx}(\dfrac{f(x)}{g(x)})=\dfrac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}$

The given function : $y=\dfrac{\ln x}{x^3}$

Differentiate both sides , we get

$\dfrac{dy}{dx}=\dfrac{d}{dx}(\dfrac{\ln x}{x^3})$

By Quotient rule , this is equal to

$=\dfrac{(\ln x)'(x^3)-(x^3)'(\lnx)}{(x^3)^2}$

$=\dfrac{(\dfrac{1}{x})(x^3)-\ln x(3x^2)}{x^6}$

[∵ $\dfrac{d}{dx}(\ln x)=\dfrac{1}{x}\ \ \&\ \dfrac{d}{dx}(x^n)=n(x)^{n-1}$ ]

$=\dfrac{x^2-3x^2\ln x}{x^6} \\\\=\dfrac{x^2(1-3\ln x)}{x^6}\\\\=\dfrac{(1-3\ln x)}{x^{6-2}}\\\\=\dfrac{1-3\ln x}{x^{4}}$

Hence, the derivative of the function will be : $\dfrac{1-3ln x}{x^{4}}$

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Question: The perimeter of a rectangle is 60 cm. Its length is 12 cm. What is its width? 1. Write an expression using x to represent the rectangle's width. 2. Solve for the width.