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

Identify Function: Identify the function to differentiate. We are given the function f(x) = (x)/(3-2x^3). We need to find its derivative, which is denoted by 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 its derivative g'(x) is (u'(x)v(x) - u(x)v'(x))/(v(x))^2. Here, u(x) = x and v(x) = 3 - 2x^3.Differentiate Functions: Differentiate u(x) and v(x). The derivative of u(x) = x with respect to x is u'(x) = 1. The derivative of v(x) = 3 - 2x^3 with respect to x is v'(x) = -6x^2.Apply Derivatives: Apply the derivatives to the quotient rule. Using the quotient rule from Step 2, we get: f'(x) = (u'(x)v(x) - u(x)v'(x))/(v(x))^2 f'(x) = (1*(3 - 2x^3) - x*(-6x^2))/((3 - 2x^3)^2)Simplify Expression: Simplify the expression. Now we simplify the numerator: f'(x) = (3 - 2x^3 + 6x^3)/((3 - 2x^3)^2) f'(x) = (3 + 4x^3)/((3 - 2x^3)^2)

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

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

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Math Problems

Algebra 2

Evaluate exponential functions

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Q. Given the function

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

, find

f^{\prime}(x)

in simplified form.

\newline

Answer:

f^{\prime}(x)=

Identify Function: Identify the function to differentiate. $\newline$ We are given the function $f(x) = \frac{x}{3-2x^3}$ . We need to find its derivative, which is denoted by $f'(x)$ .
Apply Quotient Rule: Apply the quotient rule for differentiation. The quotient rule states that if we have a function $g(x) = \frac{u(x)}{v(x)}$ , then its derivative $g'(x)$ is $\frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ . Here, $u(x) = x$ and $v(x) = 3 - 2x^3$ .
Differentiate Functions: Differentiate $u(x)$ and $v(x)$ . $\newline$ The derivative of $u(x) = x$ with respect to $x$ is $u'(x) = 1$ . $\newline$ The derivative of $v(x) = 3 - 2x^3$ with respect to $x$ is $v'(x) = -6x^2$ .
Apply Derivatives: Apply the derivatives to the quotient rule. $\newline$ Using the quotient rule from Step $2$ , we get: $\newline$ $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ $\newline$ $f'(x) = \frac{1(3 - 2x^3) - x(-6x^2)}{(3 - 2x^3)^2}$
Simplify Expression: Simplify the expression. $\newline$ Now we simplify the numerator: $\newline$ $f'(x) = \frac{3 - 2x^3 + 6x^3}{(3 - 2x^3)^2}$ $\newline$ $f'(x) = \frac{3 + 4x^3}{(3 - 2x^3)^2}$