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Natasha_Volkova [10]
3 years ago
13

Y equals the quotient of the quantity x squared plus 4 times x and the quantity x cubed minus 5.

Mathematics
1 answer:
koban [17]3 years ago
4 0

Given:

Consider the completer question is "Find the derivative \dfrac{dy}{dx} for y=\dfrac{x^2-4x}{x^3-5}."

To find:

The derivative \dfrac{dy}{dx}.

Solution:

Chain rule: \dfrac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)

Quotient rule: \dfrac{d}{dx}\dfrac{f(x)}{g(x)}=\dfrac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}

We have,

y=\dfrac{x^2-4x}{x^3-5}

Differentiate with respect to x.

\dfrac{dy}{dx}=\dfrac{d}{dx}\left(\dfrac{x^2-4x}{x^3-5}\right)

Using chain rule and quotient rule, we get

\dfrac{dy}{dx}=\dfrac{(x^3-5)\dfrac{d}{dx}(x^2-4x)-(x^2-4x)\dfrac{d}{dx}(x^3-5)}{(x^3-5)^2}

\dfrac{dy}{dx}=\dfrac{(x^3-5)(2x-4)-(x^2-4x)(3x^2)}{(x^3-5)^2}

\dfrac{dy}{dx}=\dfrac{2x^4-4x^3-10x+20-3x^4+12x^3}{(x^3-5)^2}

\dfrac{dy}{dx}=\dfrac{-x^4+8x^3-10x+20}{(x^3-5)^2}

Therefore, the required answer is \dfrac{dy}{dx}=\dfrac{-x^4+8x^3-10x+20}{(x^3-5)^2}.

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