Given the function f(x)=(x)/(5-3x^(4)), find f^(')(x) in simplified form. Answer: f^(')(x)=

Identify function: Identify the function to differentiate. We are given the function f(x) = (x)/(5 - 3x^4). 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) = 5 - 3x^4.Differentiate u and v: 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) = 5 - 3x^4 with respect to x is v'(x) = -12x^3.Apply derivatives to rule: Apply the derivatives to the quotient rule. Using the derivatives from Step 3, we substitute into the quotient rule formula: f'(x) = (1 * (5 - 3x^4) - x * (-12x^3)) / (5 - 3x^4)^2.Simplify numerator: Simplify the numerator. Simplify the expression in the numerator: f'(x) = (5 - 3x^4 + 12x^4) / (5 - 3x^4)^2.Combine like terms: Combine like terms in the numerator. Combine the x^4 terms: f'(x) = (5 + 9x^4) / (5 - 3x^4)^2.Check final expression: Check the final expression. The final expression for the derivative is f'(x) = (5 + 9x^4) / (5 - 3x^4)^2. This is the simplified form of the derivative.

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Given the function
f(x)=(x)/(5-3x^(4)), find
f^(')(x) in simplified form.
Answer:
f^(')(x)=

Given the function $f(x)=\frac{x}{5-3 x^{4}}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$

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Algebra 2

Evaluate exponential functions

Full solution

Q. Given the function

f(x)=\frac{x}{5-3 x^{4}}

, 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 $f(x) = \frac{x}{5 - 3x^4}$ . 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 $g(x) = \frac{u(x)}{v(x)}$ , then its derivative $g'(x)$ is $\frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ . Here, $u(x) = x$ and $v(x) = 5 - 3x^4$ .
Differentiate $u$ and $v$ : 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 $v(x) = 5 - 3x^4$ with respect to $x$ is $v'(x) = -12x^3$ .
Apply derivatives to rule: Apply the derivatives to the quotient rule. $\newline$ Using the derivatives from Step $3$ , we substitute into the quotient rule formula: $\newline$ $f'(x) = \frac{1 \cdot (5 - 3x^4) - x \cdot (-12x^3)}{(5 - 3x^4)^2}$ .
Simplify numerator: Simplify the numerator. $\newline$ Simplify the expression in the numerator: $\newline$ $f'(x) = \frac{5 - 3x^4 + 12x^4}{(5 - 3x^4)^2}$ .
Combine like terms: Combine like terms in the numerator. $\newline$ Combine the $x^4$ terms: $\newline$ $f'(x) = \frac{5 + 9x^4}{(5 - 3x^4)^2}$ .
Check final expression: Check the final expression. $\newline$ The final expression for the derivative is $f'(x) = \frac{5 + 9x^4}{(5 - 3x^4)^2}$ . This is the simplified form of the derivative.