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

Identify u(x) and v(x): We need to find the derivative of the function f(x) = x / (5 - 4x^4). This is a quotient, so we will use the quotient rule for differentiation, which states that if we have a function g(x) = u(x) / v(x), then g'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2.Compute derivatives u'(x) and v'(x): First, identify u(x) and v(x). Here, u(x) = x and v(x) = 5 - 4x^4. Then compute the derivatives u'(x) and v'(x). The derivative of u(x) with respect to x is u'(x) = 1. The derivative of v(x) with respect to x is v'(x) = d/dx(5) - d/dx(4x^4) = 0 - 16x^3.Apply quotient rule: Now apply the quotient rule. f'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2 = (1 * (5 - 4x^4) - x * (-16x^3)) / (5 - 4x^4)^2.Simplify numerator: Simplify the numerator of the derivative. f'(x) = (5 - 4x^4 + 16x^4) / (5 - 4x^4)^2 = (5 + 12x^4) / (5 - 4x^4)^2.Final derivative: The derivative f'(x) is now in simplified form. f'(x) = (5 + 12x^4) / (5 - 4x^4)^2.

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

Given the function $f(x)=\frac{x}{5-4 x^{4}}$ , 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

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Q. Given the function

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

, find

f^{\prime}(x)

in simplified form.

\newline

Answer:

f^{\prime}(x)=

Identify $u(x)$ and $v(x)$ : We need to find the derivative of the function $f(x) = \frac{x}{5 - 4x^4}$ . This is a quotient, so we will use the quotient rule for differentiation, which states that if we have a function $g(x) = \frac{u(x)}{v(x)}$ , then $g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ .
Compute derivatives $u'(x)$ and $v'(x)$ : First, identify $u(x)$ and $v(x)$ . Here, $u(x) = x$ and $v(x) = 5 - 4x^4$ . Then compute the derivatives $u'(x)$ and $v'(x)$ . The derivative of $u(x)$ with respect to $x$ is $v'(x)$ $0$ . The derivative of $v(x)$ with respect to $x$ is $v'(x)$ $3$ .
Apply quotient rule: Now apply the quotient rule. $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} = \frac{(1 \cdot (5 - 4x^4) - x \cdot (-16x^3))}{(5 - 4x^4)^2}$ .
Simplify numerator: Simplify the numerator of the derivative. $f'(x) = \frac{5 - 4x^4 + 16x^4}{(5 - 4x^4)^2} = \frac{5 + 12x^4}{(5 - 4x^4)^2}$ .
Final derivative: The derivative $f'(x)$ is now in simplified form. $f'(x) = \frac{5 + 12x^4}{(5 - 4x^4)^2}$ .