Given the function f(x)=(6-x)/(1+x^(3)), find f^(')(x) in simplified form. Answer: f^(')(x)=

Identify function: Identify the function to differentiate. We have the function f(x) = (6-x)/(1+x^3). We need to find its derivative, denoted as f^(')(x).Apply quotient rule: Apply the quotient rule for differentiation. The quotient rule states that if we have a function g(x) = u(x)/v(x), then g^(')(x) = (u^(')(x)v(x) - u(x)v^(')(x))/(v(x))^2. Here, u(x) = 6-x and v(x) = 1+x^3.Differentiate u and v: Differentiate u(x) and v(x). u^(')(x) = -1, since the derivative of 6 is 0 and the derivative of -x is -1. v^(')(x) = 3x^2, since the derivative of 1 is 0 and the derivative of x^3 is 3x^2.Apply quotient rule: Apply the quotient rule using the derivatives from Step 3. f^(')(x) = ((-1)(1+x^3) - (6-x)(3x^2)) / (1+x^3)^2.Expand and simplify: Expand and simplify the numerator of the derivative. f^(')(x) = (-1 - x^3 - 18x^2 + 3x^3) / (1+x^3)^2.Combine like terms: Combine like terms in the numerator. f^(')(x) = (-1 - 18x^2 + 2x^3) / (1+x^3)^2.Check for simplification: Check for any possible simplification. The numerator and denominator are already in their simplest form, so no further simplification is possible.

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Given the function
f(x)=(6-x)/(1+x^(3)), find
f^(')(x) in simplified form.
Answer:
f^(')(x)=

Given the function $f(x)=\frac{6-x}{1+x^{3}}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$

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Algebra 2

Find the vertex of the transformed function

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Q. Given the function

f(x)=\frac{6-x}{1+x^{3}}

, find

f^{\prime}(x)

in simplified form.

\newline

Answer:

f^{\prime}(x)=

Identify function: Identify the function to differentiate. $\newline$ We have the function $f(x) = \frac{6-x}{1+x^3}$ . We need to find its derivative, denoted as $f^{\prime}(x)$ .
Apply quotient rule: Apply the quotient rule for differentiation. $\newline$ The quotient rule states that if we have a function $g(x) = \frac{u(x)}{v(x)}$ , then $g^{\prime}(x) = \frac{u^{\prime}(x)v(x) - u(x)v^{\prime}(x)}{(v(x))^2}$ . Here, $u(x) = 6-x$ and $v(x) = 1+x^3$ .
Differentiate $u$ and $v$ : Differentiate $u(x)$ and $v(x)$ . $u^{\prime}(x) = -1$ , since the derivative of $6$ is $0$ and the derivative of $-x$ is $-1$ . $v^{\prime}(x) = 3x^2$ , since the derivative of $v$ $0$ is $0$ and the derivative of $v$ $2$ is $v$ $3$ .
Apply quotient rule: Apply the quotient rule using the derivatives from Step $3$ . $\newline$ $f'(x) = \frac{(-1)(1+x^3) - (6-x)(3x^2)}{(1+x^3)^2}$ .
Expand and simplify: Expand and simplify the numerator of the derivative. $f'(x) = \frac{-1 - x^3 - 18x^2 + 3x^3}{(1+x^3)^2}$ .
Combine like terms: Combine like terms in the numerator. $\newline$ $f'(x) = \frac{-1 - 18x^2 + 2x^3}{(1+x^3)^2}$ .
Check for simplification: Check for any possible simplification. $\newline$ The numerator and denominator are already in their simplest form, so no further simplification is possible.