Given the function f(x)=(x)/(5-2x^(4)), find f^(')(x) in simplified form. Answer: f^(')(x)=

Apply Quotient Rule: To find the derivative of the function f(x) = x / (5 - 2x^4), we will use the quotient rule. The quotient rule states that if we 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 Here, u(x) = x and v(x) = 5 - 2x^4.Find u'(x): First, we find the derivative of u(x) with respect to x. Since u(x) = x, the derivative u'(x) is: u'(x) = d(x)/dx = 1Find v'(x): Next, we find the derivative of v(x) with respect to x. Since v(x) = 5 - 2x^4, the derivative v'(x) is: v'(x) = d(5 - 2x^4)/dx = 0 - 2 * 4x^3 = -8x^3Apply Quotient Rule: Now we apply the quotient rule using the derivatives u'(x) and v'(x) we found: f'(x) = ((5 - 2x^4) * (1) - (x) * (-8x^3)) / (5 - 2x^4)^2Simplify Numerator: Simplify the numerator of the derivative: f'(x) = (5 - 2x^4 - (-8x^4)) / (5 - 2x^4)^2 f'(x) = (5 + 6x^4) / (5 - 2x^4)^2Final Derivative: The derivative f'(x) is now in simplified form: f'(x) = (5 + 6x^4) / (25 - 20x^4 + 4x^8)

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Given the function
f(x)=(x)/(5-2x^(4)), find
f^(')(x) in simplified form.
Answer:
f^(')(x)=

Given the function $f(x)=\frac{x}{5-2 x^{4}}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$

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Algebra 2

Find the vertex of the transformed function

Full solution

Q. Given the function

f(x)=\frac{x}{5-2 x^{4}}

, find

f^{\prime}(x)

in simplified form.

\newline

Answer:

f^{\prime}(x)=

Apply Quotient Rule: To find the derivative of the function $f(x) = \frac{x}{5 - 2x^4}$ , we will use the quotient rule. 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 $f'(x)$ is given by: $\newline$ $f'(x) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2}$ $\newline$ Here, $u(x) = x$ and $v(x) = 5 - 2x^4$ .
Find $u'(x)$ : First, we find the derivative of $u(x)$ with respect to $x$ . Since $u(x) = x$ , the derivative $u'(x)$ is: $\newline$ $u'(x) = \frac{d(x)}{dx} = 1$
Find $v'(x)$ : Next, we find the derivative of $v(x)$ with respect to $x$ . Since $v(x) = 5 - 2x^4$ , the derivative $v'(x)$ is: $\newline$ $v'(x) = \frac{d(5 - 2x^4)}{dx} = 0 - 2 \cdot 4x^3 = -8x^3$
Apply Quotient Rule: Now we apply the quotient rule using the derivatives $u'(x)$ and $v'(x)$ we found: $f'(x) = \frac{(5 - 2x^4) \cdot (1) - (x) \cdot (-8x^3)}{(5 - 2x^4)^2}$
Simplify Numerator: Simplify the numerator of the derivative: $\newline$ $f'(x) = \frac{5 - 2x^4 - (-8x^4)}{(5 - 2x^4)^2}$ $\newline$ $f'(x) = \frac{5 + 6x^4}{(5 - 2x^4)^2}$
Final Derivative: The derivative $f'(x)$ is now in simplified form: $\newline$ $f'(x) = \frac{5 + 6x^4}{25 - 20x^4 + 4x^8}$