Find (d)/(dx)([ln(x)]^(3)). Choose 1 answer: (A) 3[ln(x)]^(2) (B) 3((1)/(x))^(2) (C) (3[ln(x)]^(2))/(x) (D) ((1)/(x))^(3)

Apply Chain Rule: Apply the chain rule to differentiate [ln(x)]^(3). The chain rule states that the derivative of a composite function f(g(x)) is f'(g(x))g'(x). In this case, f(u) = u^3 and g(x) = ln(x), so we need to find the derivatives f'(u) and g'(x).Derivative of f(u): Find the derivative of f(u) = u^3 with respect to u. Using the power rule, (d/du)(u^n) = nu^(n-1), we get: (d/du)(u^3) = 3u^(3-1) = 3u^2.Derivative of g(x): Find the derivative of g(x) = ln(x) with respect to x. The derivative of ln(x) with respect to x is 1/x.Apply Chain Rule: Apply the chain rule using the derivatives from steps 2 and 3. (d/dx)([ln(x)]^(3)) = f'(g(x))g'(x) = 3[ln(x)]^2 * (1/x).Simplify Expression: Simplify the expression. (d/dx)([ln(x)]^(3)) = (3[ln(x)]^2)/(x). This matches answer choice (C).

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Find
(d)/(dx)([ln(x)]^(3)).
Choose 1 answer:
(A)
3[ln(x)]^(2)
(B)
3((1)/(x))^(2)
(C)
(3[ln(x)]^(2))/(x)
(D)
((1)/(x))^(3)

Find $\frac{d}{d x}\left([\ln (x)]^{3}\right)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $3[\ln (x)]^{2}$ $\newline$ (B) $3\left(\frac{1}{x}\right)^{2}$ $\newline$ (C) $\frac{3[\ln (x)]^{2}}{x}$ $\newline$ (D) $\left(\frac{1}{x}\right)^{3}$

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Calculus

Find derivatives of using multiple formulae

Full solution

Q. Find

\frac{d}{d x}\left([\ln (x)]^{3}\right)

\newline

Choose

1

answer:

\newline

(A)

3[\ln (x)]^{2}

\newline

(B)

3\left(\frac{1}{x}\right)^{2}

\newline

(C)

\frac{3[\ln (x)]^{2}}{x}

\newline

(D)

\left(\frac{1}{x}\right)^{3}

Apply Chain Rule: Apply the chain rule to differentiate $[\ln(x)]^{3}$ . The chain rule states that the derivative of a composite function $f(g(x))$ is $f'(g(x))g'(x)$ . In this case, $f(u) = u^{3}$ and $g(x) = \ln(x)$ , so we need to find the derivatives $f'(u)$ and $g'(x)$ .
Derivative of $f(u)$ : Find the derivative of $f(u) = u^3$ with respect to $u$ . Using the power rule, $\frac{d}{du}(u^n) = nu^{n-1}$ , we get: $\frac{d}{du}(u^3) = 3u^{3-1} = 3u^2$ .
Derivative of $g(x)$ : Find the derivative of $g(x) = \ln(x)$ with respect to $x$ . The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$ .
Apply Chain Rule: Apply the chain rule using the derivatives from steps $2$ and $3$ . $\newline$ $(\frac{d}{dx})([\ln(x)]^{3}) = f'(g(x))g'(x) = 3[\ln(x)]^2 \cdot (\frac{1}{x}).$
Simplify Expression: Simplify the expression. $\newline$ $(\frac{d}{dx})([\ln(x)]^{3}) = \frac{3[\ln(x)]^2}{x}$ . $\newline$ This matches answer choice (C).