Question

Find the derivitive of the function: y=(-5x^2 + 3x + 2) / x

Answer 1

`(d)/(dx)\lbrack(-5x^2+3x+2)/(x)\rbrack=-(5x^2+2)/(x^2)`

Explanation:

To solve this, we'll use the quotient rule for derivatives. This rule tells us that:

`(d)/(dx)\lbrack(f(x))/(g(x))\rbrack=(g(x)f^(\prime)(x)-f(x)g^(\prime)(x))/(\lbrack g(x)\rbrack^2)`

For the given expression, we'll have that:

`\begin{gathered} f(x)=-5x^2+3x+2 \\ f^(\prime)(x)=-10x+3 \\ \\ g(x)=x \\ g^(\prime)(x)=1 \\ \lbrack g(x)\rbrack^2=x^2 \end{gathered}`

This way, we'll have that:

`(d)/(dx)\lbrack(-5x^2+3x+2)/(x)\rbrack=((x)(-10x+3)-(-5x^2+3x+2)(1))/(x^2)`

Simplifying this expression,

`\begin{gathered} ((x)(-10x+3)-(-5x^(2)+3x+2)(1))/(x^(2)) \\ \\ \rightarrow(-10x^2+3x-+5x^2-3x-2)/(x^2) \\ \\ \rightarrow(-5x^2-2)/(x^2) \\ \\ \Rightarrow-(5x^2+2)/(x^2) \end{gathered}`

Thus, we can conclude that:

`(d)/(dx)\lbrack(-5x^2+3x+2)/(x)\rbrack=-(5x^2+2)/(x^2)`