Given the function y=(x)/(6-x^(2)), find (dy)/(dx) in simplified form. Answer: (dy)/(dx)=

Identify function: Identify the function to differentiate. We are given the function y = (x)/(6-x^2). We need to find its derivative with respect to x, which is denoted as (dy)/(dx).Apply quotient rule: Apply the quotient rule for differentiation. The quotient rule states that the derivative of a function in the form of u/v is given by (v(u') - u(v')) / v^2, where u' and v' are the derivatives of u and v with respect to x, respectively. Here, u = x and v = 6 - x^2. We need to find u' and v'.Differentiate u and v: Differentiate u and v with respect to x. u = x, so u' = d(x)/dx = 1. v = 6 - x^2, so v' = d(6 - x^2)/dx = 0 - 2x = -2x.Apply quotient rule: Apply the quotient rule using the derivatives from Step 3. (dy)/(dx) = ((6 - x^2)(1) - (x)(-2x)) / (6 - x^2)^2Simplify expression: Simplify the expression. (dy)/(dx) = (6 - x^2 + 2x^2) / (6 - x^2)^2 (dy)/(dx) = (6 + x^2) / (6 - x^2)^2

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Given the function
y=(x)/(6-x^(2)), find
(dy)/(dx) in simplified form.
Answer:
(dy)/(dx)=

Given the function $y=\frac{x}{6-x^{2}}$ , find $\frac{d y}{d x}$ in simplified form. $\newline$ Answer: $\frac{d y}{d x}=$

View step-by-step help

Full solution

Q. Given the function

y=\frac{x}{6-x^{2}}

, find

\frac{d y}{d x}

in simplified form.

\newline

Answer:

\frac{d y}{d x}=

Identify function: Identify the function to differentiate. $\newline$ We are given the function $y = \frac{x}{6-x^2}$ . We need to find its derivative with respect to $x$ , which is denoted as $\frac{dy}{dx}$ .
Apply quotient rule: Apply the quotient rule for differentiation. $\newline$ The quotient rule states that the derivative of a function in the form of $\frac{u}{v}$ is given by $\frac{v(u') - u(v')}{v^2}$ , where $u'$ and $v'$ are the derivatives of $u$ and $v$ with respect to $x$ , respectively. $\newline$ Here, $u = x$ and $v = 6 - x^2$ . We need to find $u'$ and $v'$ .
Differentiate $u$ and $v$ : Differentiate $u$ and $v$ with respect to $x$ .
$u = x$ , so $u' = \frac{d(x)}{dx} = 1$ .
$v = 6 - x^2$ , so $v' = \frac{d(6 - x^2)}{dx} = 0 - 2x = -2x$ .
Apply quotient rule: Apply the quotient rule using the derivatives from Step $3$ . $\frac{dy}{dx} = \frac{(6 - x^2)(1) - (x)(-2x)}{(6 - x^2)^2}$
Simplify expression: Simplify the expression. $\newline$ $(dy)/(dx) = (6 - x^2 + 2x^2) / (6 - x^2)^2$ $\newline$ $(dy)/(dx) = (6 + x^2) / (6 - x^2)^2$