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Given the function
f(x)=(x)/(3-4x^(3)), find
f^(')(x) in simplified form.
Answer:
f^(')(x)=

Given the function $f(x)=\frac{x}{3-4 x^{3}}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$

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Write a formula for an arithmetic sequence

Full solution

Q. Given the function

f(x)=\frac{x}{3-4 x^{3}}

, find

f^{\prime}(x)

in simplified form.

\newline

Answer:

f^{\prime}(x)=

Identify Function: Identify the function that needs to be differentiated. $\newline$ The function given is $f(x) = \frac{x}{3-4x^3}$ . We need to find its derivative, which is denoted by $f'(x)$ .
Apply Quotient Rule: Apply the quotient rule for differentiation. $\newline$ The quotient rule states that if we have a function that is the quotient of two functions, $\frac{u(x)}{v(x)}$ , then its derivative is given by $\frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}$ . Here, $u(x) = x$ and $v(x) = 3 - 4x^3$ .
Differentiate $u(x)$ : Differentiate $u(x)$ with respect to $x$ . The derivative of $u(x) = x$ with respect to $x$ is $u'(x) = 1$ .
Differentiate $v(x)$ : Differentiate $v(x)$ with respect to $x$ . The derivative of $v(x) = 3 - 4x^3$ with respect to $x$ is $v'(x) = -12x^2$ .
Substitute into Formula: Substitute $u(x)$ , $u'(x)$ , $v(x)$ , and $v'(x)$ into the quotient rule formula. $\newline$ Using the quotient rule, we get $f'(x) = \frac{(3 - 4x^3)(1) - (x)(-12x^2)}{(3 - 4x^3)^2}$ .
Simplify Expression: Simplify the expression. $\newline$ Simplify the numerator: $(3 - 4x^3)(1) - (x)(-12x^2) = 3 - 4x^3 + 12x^3$ . $\newline$ Combine like terms: $3 + 8x^3$ . $\newline$ Now, $f'(x) = \frac{3 + 8x^3}{(3 - 4x^3)^2}$ .

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