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

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 that is the quotient of two functions, u(x)/v(x), then its derivative is given by (v(x)u'(x) - u(x)v'(x))/(v(x))^2. Here, u(x) = x^2 and v(x) = ln(x).Find u(x): First, we need to find the derivative of u(x) = x^2. The derivative of x^2 with respect to x is 2x.Find v(x): Next, we need to 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. The derivative of f(x) is given by: f'(x) = ((ln(x))(2x) - (x^2)(1/x))/(ln(x))^2Simplify Numerator: Simplify the expression by performing the multiplication and division in the numerator: f'(x) = (2x*ln(x) - x^2/x)/(ln(x))^2Further Simplify: Further simplify the expression by canceling the x in the term x^2/x: f'(x) = (2x*ln(x) - x)/(ln(x))^2Final Derivative: Now we have the derivative in its simplified form: f'(x) = (2x*ln(x) - x)/(ln(x))^2 This matches answer choice (A).

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

Let $f(x)=\frac{x^{2}}{\ln (x)}$ . $\newline$ Find $f^{\prime}(x)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{2 x \ln (x)-x}{(\ln (x))^{2}}$ $\newline$ (B) $2 x \ln (x)+x$ $\newline$ (C) $2 x-\frac{1}{x}$ $\newline$ (D) $2 x^{2}$

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Math Problems

Calculus

Find derivatives of using multiple formulae

Full solution

Q. Let

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

\newline

Find

f^{\prime}(x)

\newline

Choose

1

answer:

\newline

(A)

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

\newline

(B)

2 x \ln (x)+x

\newline

(C)

2 x-\frac{1}{x}

\newline

(D)

2 x^{2}

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 that is the quotient of two functions, $\frac{u(x)}{v(x)}$ , then its derivative is given by $\frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}$ . Here, $u(x) = x^2$ and $v(x) = \ln(x)$ .
Find $u(x)$ : First, we need to find the derivative of $u(x) = x^2$ . The derivative of $x^2$ with respect to $x$ is $2x$ .
Find $v(x)$ : Next, we need to 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. The derivative of $f(x)$ is given by: $\newline$ $f'(x) = \frac{(\ln(x))(2x) - (x^2)(\frac{1}{x})}{(\ln(x))^2}$
Simplify Numerator: Simplify the expression by performing the multiplication and division in the numerator: $f'(x) = \frac{2x\cdot\ln(x) - \frac{x^2}{x}}{(\ln(x))^2}$
Further Simplify: Further simplify the expression by canceling the $x$ in the term $\frac{x^2}{x}$ : $f'(x) = \frac{2x\cdot\ln(x) - x}{(\ln(x))^2}$
Final Derivative: Now we have the derivative in its simplified form: $\newline$ $f'(x) = \frac{2x\ln(x) - x}{(\ln(x))^2}$ $\newline$ This matches answer choice (A).