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

Identify function: Identify the function to differentiate. f(x) = (x^3 - 1) / (6 + x^3) We need to find the derivative of this function with respect to x, 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) = x^3 - 1 and v(x) = 6 + x^3.Differentiate u and v: Differentiate u(x) and v(x) with respect to x. u'(x) = d/dx (x^3 - 1) = 3x^2 v'(x) = d/dx (6 + x^3) = 3x^2Apply quotient rule: Apply the quotient rule using the derivatives from the previous step. f'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2 f'(x) = (3x^2(6 + x^3) - (x^3 - 1)(3x^2)) / (6 + x^3)^2Expand derivative: Expand the numerator of the derivative. f'(x) = (18x^2 + 3x^5 - 3x^5 + 3x^2) / (6 + x^3)^2 Simplify the terms in the numerator. f'(x) = (18x^2 + 3x^2) / (6 + x^3)^2 f'(x) = 21x^2 / (6 + x^3)^2Check for simplification: Check for any possible simplification or common factors. There are no common factors to cancel out, and the expression is already in its simplest form.

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

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

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Math Problems

Algebra 1

Write variable expressions for arithmetic sequences

Full solution

Q. Given the function

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

, find

f^{\prime}(x)

in simplified form.

\newline

Answer:

f^{\prime}(x)=

Identify function: Identify the function to differentiate. $\newline$ $f(x) = \frac{x^3 - 1}{6 + x^3}$ $\newline$ We need to find the derivative of this function with respect to $x$ , denoted as $f'(x)$ .
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 $g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ . $\newline$ Here, $u(x) = x^3 - 1$ and $v(x) = 6 + x^3$ .
Differentiate $u$ and $v$ : Differentiate $u(x)$ and $v(x)$ with respect to $x$ .
$u'(x) = \frac{d}{dx} (x^3 - 1) = 3x^2$
$v'(x) = \frac{d}{dx} (6 + x^3) = 3x^2$
Apply quotient rule: Apply the quotient rule using the derivatives from the previous step. $\newline$ $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ $\newline$ $f'(x) = \frac{(3x^2(6 + x^3) - (x^3 - 1)(3x^2))}{(6 + x^3)^2}$
Expand derivative: Expand the numerator of the derivative. $\newline$ $f'(x) = \frac{18x^2 + 3x^5 - 3x^5 + 3x^2}{(6 + x^3)^2}$ $\newline$ Simplify the terms in the numerator. $\newline$ $f'(x) = \frac{18x^2 + 3x^2}{(6 + x^3)^2}$ $\newline$ $f'(x) = \frac{21x^2}{(6 + x^3)^2}$
Check for simplification: Check for any possible simplification or common factors. There are no common factors to cancel out, and the expression is already in its simplest form.