Given the function f(x)=(1+2x^(3))/(5-2x^(3)), find f^(')(x) in simplified form. Answer: f^(')(x)=

Identify Functions: To find the derivative of the function f(x) = (1+2x^3)/(5-2x^3), we will use the quotient rule. The quotient rule states that if you have a function that is the quotient of two functions, u(x)/v(x), then its derivative f'(x) is given by: f'(x) = (v(x)u'(x) - u(x)v'(x)) / (v(x))^2 Let's identify u(x) and v(x) for our function: u(x) = 1+2x^3 and v(x) = 5-2x^3Find Derivatives: Next, we need to find the derivatives of u(x) and v(x), which are u'(x) and v'(x) respectively. The derivative of u(x) = 1+2x^3 with respect to x is u'(x) = 0 + 6x^2. The derivative of v(x) = 5-2x^3 with respect to x is v'(x) = 0 - 6x^2.Apply Quotient Rule: Now we can apply the quotient rule using the derivatives we found: f'(x) = ((5-2x^3)(6x^2) - (1+2x^3)(-6x^2)) / (5-2x^3)^2Simplify Numerator: Simplify the numerator of the derivative: f'(x) = (30x^2 - 12x^5 + 6x^2 + 12x^5) / (5-2x^3)^2 Combine like terms in the numerator: f'(x) = (30x^2 + 6x^2) / (5-2x^3)^2 f'(x) = 36x^2 / (5-2x^3)^2Combine Like Terms: The derivative f'(x) is now in simplified form: f'(x) = 36x^2 / (5-2x^3)^2 This is the final answer.

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Given the function
f(x)=(1+2x^(3))/(5-2x^(3)), find
f^(')(x) in simplified form.
Answer:
f^(')(x)=

Given the function $f(x)=\frac{1+2 x^{3}}{5-2 x^{3}}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$

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Algebra 2

Csc, sec, and cot of special angles

Full solution

Q. Given the function

f(x)=\frac{1+2 x^{3}}{5-2 x^{3}}

, find

f^{\prime}(x)

in simplified form.

\newline

Answer:

f^{\prime}(x)=

Identify Functions: To find the derivative of the function $f(x) = \frac{1+2x^3}{5-2x^3}$ , we will use the quotient rule. The quotient rule states that if you have a function that is the quotient of two functions, $\frac{u(x)}{v(x)}$ , then its derivative $f'(x)$ is given by: $\newline$ $f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}$ $\newline$ Let's identify $u(x)$ and $v(x)$ for our function: $\newline$ $u(x) = 1+2x^3$ and $v(x) = 5-2x^3$
Find Derivatives: Next, we need to find the derivatives of $u(x)$ and $v(x)$ , which are $u'(x)$ and $v'(x)$ respectively. $\newline$ The derivative of $u(x) = 1+2x^3$ with respect to $x$ is $u'(x) = 0 + 6x^2$ . $\newline$ The derivative of $v(x) = 5-2x^3$ with respect to $x$ is $v'(x) = 0 - 6x^2$ .
Apply Quotient Rule: Now we can apply the quotient rule using the derivatives we found: $f'(x) = \frac{(5-2x^3)(6x^2) - (1+2x^3)(-6x^2)}{(5-2x^3)^2}$
Simplify Numerator: Simplify the numerator of the derivative: $\newline$ $f'(x) = \frac{30x^2 - 12x^5 + 6x^2 + 12x^5}{(5-2x^3)^2}$ $\newline$ Combine like terms in the numerator: $\newline$ $f'(x) = \frac{30x^2 + 6x^2}{(5-2x^3)^2}$ $\newline$ $f'(x) = \frac{36x^2}{(5-2x^3)^2}$
Combine Like Terms: The derivative $f'(x)$ is now in simplified form: $f'(x) = \frac{36x^2}{(5-2x^3)^2}$ This is the final answer.