Let f(x)=(x^(2))/(ln(x)). Find f^(')(x). Choose 1 answer: (A) 2x^(2) (B) (2x ln(x)-x)/((ln(x))^(2)) (C) 2x ln(x)+x (D) 2x-(1)/(x)

Use Quotient Rule: To find the derivative of the function f(x) = (x^2)/(ln(x)), we will use the quotient rule, 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. Here, u(x) = x^2 and v(x) = ln(x).Find u'(x): First, we find the derivative of u(x) = x^2. The derivative of x^2 with respect to x is 2x.Find v'(x): Next, we find the derivative of v(x) = ln(x). The derivative of ln(x) with respect to x is 1/x.Apply Quotient Rule: Now we apply the quotient rule. We have u'(x) = 2x and v'(x) = 1/x. Plugging these into the quotient rule formula, we get: f'(x) = (2x * ln(x) - x^2 * (1/x)) / (ln(x))^2.Simplify Numerator: Simplify the expression inside the numerator of the derivative: f'(x) = (2x * ln(x) - x) / (ln(x))^2.Match Answer Choice: We can see that the expression (2x * ln(x) - x) / (ln(x))^2 matches answer choice (B), which is (2x ln(x) - x) / (ln(x))^2.

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Math Problems

Algebra 2

Sum of finite series starts from 1

Full solution

Q. Let

f(x)=\frac{x^{2}}{\ln (x)}

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Find

f^{\prime}(x)

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Choose

1

answer:

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(A)

2 x^{2}

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(B)

\frac{2 x \ln (x)-x}{(\ln (x))^{2}}

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(C)

2 x \ln (x)+x

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(D)

2 x-\frac{1}{x}

Use Quotient Rule: To find the derivative of the function $f(x) = \frac{x^2}{\ln(x)}$ , we will use the quotient rule, which states that if we have a function $g(x) = \frac{u(x)}{v(x)}$ , then $g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$ . Here, $u(x) = x^2$ and $v(x) = \ln(x)$ .
Find $u'(x)$ : First, we find the derivative of $u(x) = x^2$ . The derivative of $x^2$ with respect to $x$ is $2x$ .
Find $v'(x)$ : Next, we find the derivative of $v(x) = \ln(x)$ . The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$ .
Apply Quotient Rule: Now we apply the quotient rule. We have $u'(x) = 2x$ and $v'(x) = \frac{1}{x}$ . Plugging these into the quotient rule formula, we get: $\newline$ $f'(x) = \frac{(2x \cdot \ln(x) - x^2 \cdot (\frac{1}{x}))}{(\ln(x))^2}.$
Simplify Numerator: Simplify the expression inside the numerator of the derivative: $f'(x) = \frac{2x \cdot \ln(x) - x}{(\ln(x))^2}$ .
Match Answer Choice: We can see that the expression $(2x \cdot \ln(x) - x) / (\ln(x))^2$ matches answer choice (B), which is $(2x \ln(x) - x) / (\ln(x))^2$ .