Given the function f(x)=(3-x^(2))/(1+5x^(2)), find f^(')(x) in simplified form. Answer: f^(')(x)=

Identify function: Identify the function to differentiate. We are given the function f(x) = (3 - x^2) / (1 + 5x^2). We need to find its derivative, denoted as 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) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2. Here, u(x) = 3 - x^2 and v(x) = 1 + 5x^2.Differentiate numerator: Differentiate the numerator u(x) = 3 - x^2. The derivative of u(x) with respect to x is u'(x) = -2x.Differentiate denominator: Differentiate the denominator v(x) = 1 + 5x^2. The derivative of v(x) with respect to x is v'(x) = 10x.Apply quotient rule: Apply the quotient rule using the derivatives from steps 3 and 4. f'(x) = (u'(x)v(x) - u(x)v'(x)) / (v(x))^2 = ((-2x)(1 + 5x^2) - (3 - x^2)(10x)) / (1 + 5x^2)^2Expand and simplify: Expand the numerator and simplify. f'(x) = (-2x - 10x^3 - 30x + 10x^3) / (1 + 5x^2)^2 = (-2x - 30x) / (1 + 5x^2)^2 = -32x / (1 + 5x^2)^2Check for simplification: Check for any possible simplification. The expression -32x / (1 + 5x^2)^2 is already in its simplest form.

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Given the function
f(x)=(3-x^(2))/(1+5x^(2)), find
f^(')(x) in simplified form.
Answer:
f^(')(x)=

Given the function $f(x)=\frac{3-x^{2}}{1+5 x^{2}}$ , find $f^{\prime}(x)$ in simplified form. $\newline$ Answer: $f^{\prime}(x)=$

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Algebra 1

Transformations of quadratic functions

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Q. Given the function

f(x)=\frac{3-x^{2}}{1+5 x^{2}}

, 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 $f(x) = \frac{3 - x^2}{1 + 5x^2}$ . We need to find its derivative, denoted as $f'(x)$ .
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 $g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ . Here, $u(x) = 3 - x^2$ and $v(x) = 1 + 5x^2$ .
Differentiate numerator: Differentiate the numerator $u(x) = 3 - x^2$ . The derivative of $u(x)$ with respect to $x$ is $u'(x) = -2x$ .
Differentiate denominator: Differentiate the denominator $v(x) = 1 + 5x^2$ . The derivative of $v(x)$ with respect to $x$ is $v'(x) = 10x$ .
Apply quotient rule: Apply the quotient rule using the derivatives from steps $3$ and $4$ . $\newline$ $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ $\newline$ = $\frac{((-2x)(1 + 5x^2) - (3 - x^2)(10x))}{(1 + 5x^2)^2}$
Expand and simplify: Expand the numerator and simplify. $\newline$ $f'(x) = \frac{-2x - 10x^3 - 30x + 10x^3}{(1 + 5x^2)^2}$ $\newline$ $= \frac{-2x - 30x}{(1 + 5x^2)^2}$ $\newline$ $= \frac{-32x}{(1 + 5x^2)^2}$
Check for simplification: Check for any possible simplification. $\newline$ The expression $-\frac{32x}{(1 + 5x^2)^2}$ is already in its simplest form.