QuestionDifferentiate $y=\frac{6 x^{4}-x+5}{3 x^{2}}$ and express the answer with positive exponents.

Studdy Solution

STEP 1

Assumptions1. The function is $y=\frac{6 x^{4}-x+5}{3 x^{}}$ . We are asked to find the derivative of this function3. We will use the quotient rule for differentiation which states that the derivative of $\frac{u}{v}$ is \frac{vu'-uv'}{v^} where $u$ and $v$ are functions of $x$ , and $u'$ and $v'$ are their respective derivatives.

STEP 2

First, we identify the functions $u$ and $v$ in our equation. Here, $u$ is the numerator and $v$ is the denominator.
$u =6x^4 - x +5$ $v =x^2$

STEP 3

Next, we find the derivatives of $u$ and $v$ .
For $u$ , we apply the power rule which states that the derivative of $x^n$ is $nx^{n-1}$ .
$u' = \frac{d}{dx}(6x^ - x +5) =24x^3 -1$ For $v$ , we again apply the power rule.
$v' = \frac{d}{dx}(3x^2) =6x$

STEP 4

Now, we can apply the quotient rule to find the derivative of $y$ .
$y' = \frac{vu'-uv'}{v^2}$

STEP 5

Substitute the values of $u$ , $u'$ , $v$ , and $v'$ into the formula.
$y' = \frac{(3x^2)(24x^3 -1) - (x^4 - x +5)(x)}{(3x^2)^2}$

STEP 6

implify the numerator and denominator separately.
$y' = \frac{72x^5 -3x^2 -36x^5 +6x^2 -30x}{9x^4}$

STEP 7

Combine like terms in the numerator.
$y' = \frac{36x^5 +3x^2 -30x}{9x^4}$

STEP 8

Finally, we simplify the expression by dividing each term in the numerator by $x^4$ .
$y' =4x - \frac{1}{3x^2} - \frac{10}{3x^3}$ The derivative of $y=\frac{6 x^{4}-x+5}{3 x^{2}}$ is $y'=4x - \frac{1}{3x^2} - \frac{10}{3x^3}$ .

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