Question Find the derivative of the function $g(x)=\frac{4x+3}{x^2}$ .

Studdy Solution

STEP 1

Assumptions1. We are given the function $g(x)=\frac{4x+3}{x^{}}$ . We are asked to find the derivative of this function, denoted as $g'(x)$ 3. 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 need to identify the functions $u$ and $v$ in our given function. In our case, $u =4x +$ and $v = x^2$ .

STEP 3

Next, we need to find the derivatives of $u$ and $v$ . The derivative of $u =x +3$ is $u' =$ , and the derivative of $v = x^2$ is $v' =2x$ .

STEP 4

Now we can use the quotient rule to find the derivative of $g(x)$ . $g'(x) = \frac{v \cdot u' - u \cdot v'}{v^2}$

STEP 5

Substitute the values of $u$ , $v$ , $u'$ , and $v'$ into the formula.
$g'(x) = \frac{x^2 \cdot4 - (4x +3) \cdot2x}{(x^2)^2}$

STEP 6

implify the numerator and the denominator.
$g'(x) = \frac{4x^2 -8x^2 -6x}{x^4}$

STEP 7

Further simplify the expression.
$g'(x) = \frac{-4x^2 -6x}{x^4}$

STEP 8

Finally, we can simplify the expression by dividing each term in the numerator by $x^4$ .
$g'(x) = -4x^{-2} -6x^{-3}$ So, the derivative of the function $g(x)=\frac{4x+3}{x^{2}}$ is $g'(x) = -4x^{-2} -6x^{-3}$ .

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Question Find the derivative of the function $g(x)=\frac{4x+3}{x^2}$ .

Studdy Solution

STEP 1

STEP 2

First, we need to identify the functions $u$ and $v$ in our given function. In our case, $u =4x +$ and $v = x^2$ .

STEP 3

Next, we need to find the derivatives of $u$ and $v$ . The derivative of $u =x +3$ is $u' =$ , and the derivative of $v = x^2$ is $v' =2x$ .

STEP 4

Now we can use the quotient rule to find the derivative of $g(x)$ . $g'(x) = \frac{v \cdot u' - u \cdot v'}{v^2}$

STEP 5

Substitute the values of $u$ , $v$ , $u'$ , and $v'$ into the formula.
$g'(x) = \frac{x^2 \cdot4 - (4x +3) \cdot2x}{(x^2)^2}$

STEP 6

implify the numerator and the denominator.
$g'(x) = \frac{4x^2 -8x^2 -6x}{x^4}$

STEP 7

Further simplify the expression.
$g'(x) = \frac{-4x^2 -6x}{x^4}$

STEP 8

Finally, we can simplify the expression by dividing each term in the numerator by $x^4$ .
$g'(x) = -4x^{-2} -6x^{-3}$ So, the derivative of the function $g(x)=\frac{4x+3}{x^{2}}$ is $g'(x) = -4x^{-2} -6x^{-3}$ .

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Question Find the derivative of the function g(x)=4x+3x2g(x)=\frac{4x+3}{x^2}g(x)=x24x+3​.

Studdy Solution

STEP 1

STEP 2

First, we need to identify the functions uuu and vvv in our given function. In our case, u=4x+u =4x +u=4x+ and v=x2v = x^2v=x2.

STEP 3

Next, we need to find the derivatives of uuu and vvv. The derivative of u=x+3u =x +3u=x+3 is u′=u' =u′=, and the derivative of v=x2v = x^2v=x2 is v′=2xv' =2xv′=2x.

STEP 4

Now we can use the quotient rule to find the derivative of g(x)g(x)g(x).g′(x)=v⋅u′−u⋅v′v2g'(x) = \frac{v \cdot u' - u \cdot v'}{v^2}g′(x)=v2v⋅u′−u⋅v′​

STEP 5

Substitute the values of uuu, vvv, u′u'u′, and v′v'v′ into the formula.g′(x)=x2⋅4−(4x+3)⋅2x(x2)2g'(x) = \frac{x^2 \cdot4 - (4x +3) \cdot2x}{(x^2)^2}g′(x)=(x2)2x2⋅4−(4x+3)⋅2x​

STEP 6

implify the numerator and the denominator.g′(x)=4x2−8x2−6xx4g'(x) = \frac{4x^2 -8x^2 -6x}{x^4}g′(x)=x44x2−8x2−6x​

STEP 7

Further simplify the expression.g′(x)=−4x2−6xx4g'(x) = \frac{-4x^2 -6x}{x^4}g′(x)=x4−4x2−6x​

STEP 8

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Question Find the derivative of the function $g(x)=\frac{4x+3}{x^2}$ .

First, we need to identify the functions $u$ and $v$ in our given function. In our case, $u =4x +$ and $v = x^2$ .

Next, we need to find the derivatives of $u$ and $v$ . The derivative of $u =x +3$ is $u' =$ , and the derivative of $v = x^2$ is $v' =2x$ .

Now we can use the quotient rule to find the derivative of $g(x)$ . $g'(x) = \frac{v \cdot u' - u \cdot v'}{v^2}$

Substitute the values of $u$ , $v$ , $u'$ , and $v'$ into the formula.
$g'(x) = \frac{x^2 \cdot4 - (4x +3) \cdot2x}{(x^2)^2}$

implify the numerator and the denominator.
$g'(x) = \frac{4x^2 -8x^2 -6x}{x^4}$

Further simplify the expression.
$g'(x) = \frac{-4x^2 -6x}{x^4}$