Question Use the quotient rule to evaluate h^(')(a) for the given function h(x) and a. h(x)=(3x^(2))/(x^(2)+9x+9)quad a=-3

Identify u(x) and v(x): To find h'(a), we first need to find the derivative of h(x) using the quotient rule. The quotient rule states that if we have a function h(x) = u(x) / v(x), then h'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2. Let's identify u(x) and v(x) for our function. u(x) = 3x^2 and v(x) = x^2 + 9x + 9.Find Derivatives of u(x) and v(x): Next, we need to find the derivatives of u(x) and v(x). The derivative of u(x) with respect to x is u'(x) = d/dx [3x^2] = 6x. The derivative of v(x) with respect to x is v'(x) = d/dx [x^2 + 9x + 9] = 2x + 9.Apply Quotient Rule for h'(x): Now we apply the quotient rule to find h'(x). Using the derivatives we found: h'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2 = (6x(x^2 + 9x + 9) - 3x^2(2x + 9)) / (x^2 + 9x + 9)^2.Simplify Numerator of h'(x): We simplify the numerator of h'(x): 6x(x^2 + 9x + 9) - 3x^2(2x + 9) = 6x^3 + 54x^2 + 54x - 6x^3 - 27x^2 = 27x^2 + 54x.Substitute x with a: Now we have h'(x) = (27x^2 + 54x) / (x^2 + 9x + 9)^2. To find h'(a) when a = -3, we substitute x with -3: h'(-3) = (27(-3)^2 + 54(-3)) / ((-3)^2 + 9(-3) + 9)^2 = (27(9) - 162) / (9 - 27 + 9)^2 = (243 - 162) / (-9)^2.Simplify Final Expression: Simplify the expression: h'(-3) = 81 / 81 = 1.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Question
Use the quotient rule to evaluate
h^(')(a) for the given function
h(x) and
a.

h(x)=(3x^(2))/(x^(2)+9x+9)quad a=-3

Question $\newline$ Use the quotient rule to evaluate $h^{\prime}(a)$ for the given function $h(x)$ and $a$ . $\newline$ $h(x)=\frac{3 x^{2}}{x^{2}+9 x+9} \quad a=-3$

View step-by-step help

Home

Math Problems

Algebra 1

Transformations of quadratic functions

Full solution

Q. Question

\newline

Use the quotient rule to evaluate

h^{\prime}(a)

for the given function

h(x)

and

a

\newline

h(x)=\frac{3 x^{2}}{x^{2}+9 x+9} \quad a=-3

Identify $u(x)$ and $v(x)$ : To find $h'(a)$ , we first need to find the derivative of $h(x)$ using the quotient rule. The quotient rule states that if we have a function $h(x) = \frac{u(x)}{v(x)}$ , then $h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ . Let's identify $u(x)$ and $v(x)$ for our function. $\newline$ $u(x) = 3x^2$ and $v(x) = x^2 + 9x + 9$ .
Find Derivatives of $u(x)$ and $v(x)$ : Next, we need to find the derivatives of $u(x)$ and $v(x)$ . The derivative of $u(x)$ with respect to $x$ is $u'(x) = \frac{d}{dx} [3x^2] = 6x$ . The derivative of $v(x)$ with respect to $x$ is $v'(x) = \frac{d}{dx} [x^2 + 9x + 9] = 2x + 9$ .
Apply Quotient Rule for $h'(x)$ : Now we apply the quotient rule to find $h'(x)$ . Using the derivatives we found: $\newline$ $h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} = \frac{6x(x^2 + 9x + 9) - 3x^2(2x + 9)}{(x^2 + 9x + 9)^2}.$
Simplify Numerator of $h'(x)$ : We simplify the numerator of $h'(x)$ : $6x(x^2 + 9x + 9) - 3x^2(2x + 9) = 6x^3 + 54x^2 + 54x - 6x^3 - 27x^2$ $= 27x^2 + 54x.$
Substitute $x$ with $a$ : Now we have $h'(x) = \frac{27x^2 + 54x}{(x^2 + 9x + 9)^2}$ . To find $h'(a)$ when $a = -3$ , we substitute $x$ with $-3$ :
$h'(-3) = \frac{27(-3)^2 + 54(-3)}{((-3)^2 + 9(-3) + 9)^2}$
$\phantom{h'(-3)} = \frac{27(9) - 162}{(9 - 27 + 9)^2}$
$\phantom{h'(-3)} = \frac{243 - 162}{(-9)^2}$ .
Simplify Final Expression: Simplify the expression: $\newline$ $h'(-3) = \frac{81}{81}$ $\newline$ = $1$ .