Given the function f(x)=4(-x^(2)-9x+3)^(5), find f^(')(x) in any form.

Identify Functions: We need to find the derivative of the function f(x)=4(-x^2-9x+3)^5. This requires the use of the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.Find Derivatives: First, let's identify the outer function and the inner function. The outer function is g(u)=4u^5 and the inner function is u(x)=-x^2-9x+3. We will need to find g'(u) and u'(x).Apply Chain Rule: The derivative of the outer function g(u)=4u^5 with respect to u is g'(u)=20u^4.Simplify Expression: The derivative of the inner function u(x)=-x^2-9x+3 with respect to x is u'(x)=-2x-9.Final Derivative: Now we apply the chain rule. The derivative of f(x) with respect to x is f'(x)=g'(u(x)) * u'(x). Substituting the derivatives we found, we get f'(x)=20(-x^2-9x+3)^4 * (-2x-9).Simplify the expression for f'(x) by distributing the 20 into the term (-2x-9). This gives us f'(x)=20 * (-2x-9) * (-x^2-9x+3)^4.The final form of the derivative is f'(x)=-40x(-x^2-9x+3)^4 - 180(-x^2-9x+3)^4. This is the derivative of the function f(x) in any form.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Given the function $\newline$ $f(x)=4(-x^{2}-9x+3)^{5},$ find $\newline$ $f^{\prime}(x)$ in any form.

View step-by-step help

Home

Math Problems

Algebra 2

Find the vertex of the transformed function

Full solution

Q. Given the function

\newline

f(x)=4(-x^{2}-9x+3)^{5},

find

\newline

f^{\prime}(x)

in any form.

Identify Functions: We need to find the derivative of the function $f(x)=4(-x^2-9x+3)^5$ . This requires the use of the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Find Derivatives: First, let's identify the outer function and the inner function. The outer function is $g(u)=4u^5$ and the inner function is $u(x)=-x^2-9x+3$ . We will need to find $g'(u)$ and $u'(x)$ .
Apply Chain Rule: The derivative of the outer function $g(u)=4u^5$ with respect to $u$ is $g'(u)=20u^4$ .
Simplify Expression: The derivative of the inner function $u(x)=-x^2-9x+3$ with respect to $x$ is $u'(x)=-2x-9$ .
Final Derivative: Now we apply the chain rule. The derivative of $f(x)$ with respect to $x$ is $f'(x)=g'(u(x)) \cdot u'(x)$ . Substituting the derivatives we found, we get $f'(x)=20(-x^2-9x+3)^4 \cdot (-2x-9)$ .
Final Derivative: Now we apply the chain rule. The derivative of $f(x)$ with respect to $x$ is $f'(x)=g'(u(x)) \cdot u'(x)$ . Substituting the derivatives we found, we get $f'(x)=20(-x^2-9x+3)^4 \cdot (-2x-9)$ .Simplify the expression for $f'(x)$ by distributing the $20$ into the term $(-2x-9)$ . This gives us $f'(x)=20 \cdot (-2x-9) \cdot (-x^2-9x+3)^4$ .
Final Derivative: Now we apply the chain rule. The derivative of $f(x)$ with respect to $x$ is $f'(x)=g'(u(x)) \cdot u'(x)$ . Substituting the derivatives we found, we get $f'(x)=20(-x^2-9x+3)^4 \cdot (-2x-9)$ .Simplify the expression for $f'(x)$ by distributing the $20$ into the term $(-2x-9)$ . This gives us $f'(x)=20 \cdot (-2x-9) \cdot (-x^2-9x+3)^4$ .The final form of the derivative is $f'(x)=-40x(-x^2-9x+3)^4 - 180(-x^2-9x+3)^4$ . This is the derivative of the function $f(x)$ in any form.