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Find
(d^(2))/(dx^(2))[5ln(x^(5))]

Find $\frac{d^{2}}{d x^{2}}\left[5 \ln \left(x^{5}\right)\right]$

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Find indefinite integrals using the substitution

Full solution

Q. Find

\frac{d^{2}}{d x^{2}}\left[5 \ln \left(x^{5}\right)\right]

Differentiate Function: Differentiate the function $5\ln(x^5)$ with respect to $x$ for the first time. $\newline$ Use the chain rule for differentiation: $\frac{d}{dx}[\ln(u)] = \frac{1}{u}\frac{du}{dx}$ , where $u = x^5$ . $\newline$ First derivative: $\frac{d}{dx}[5\ln(x^5)] = 5 \cdot \frac{d}{dx}[\ln(x^5)] = 5 \cdot \frac{1}{x^5} \cdot \frac{d}{dx}[x^5] = 5 \cdot \frac{1}{x^5} \cdot 5x^4 = \frac{25}{x}$ .
Use Chain Rule: Differentiate the result from Step $1$ with respect to $x$ for the second time to find the second derivative. $\newline$ Second derivative: $\frac{d^2}{dx^2}[5\ln(x^5)] = \frac{d}{dx}[\frac{25}{x}] = -\frac{25}{x^2}.$

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