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Let
h(x)=log(x).
Note: Here, we are referring to log base 10.
Find
h^('')(x).
Choose 1 answer:
(A)
-(1)/(x^(2))
(B)
-(log(x))/(ln(10))
(C)
-(1)/(x^(2)ln(10))
(D)
log(x)

Let $h(x)=\log (x)$ . $\newline$ Note: Here, we are referring to log base $10$ . $\newline$ Find $h^{\prime \prime}(x)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $-\frac{1}{x^{2}}$ $\newline$ (B) $-\frac{\log (x)}{\ln (10)}$ $\newline$ (C) $-\frac{1}{x^{2} \ln (10)}$ $\newline$ (D) $\log (x)$

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Properties of logarithms: mixed review

Full solution

Q. Let

h(x)=\log (x)

.

\newline

Note: Here, we are referring to log base

10

.

\newline

Find

h^{\prime \prime}(x)

.

\newline

Choose

1

answer:

\newline

(A)

-\frac{1}{x^{2}}

\newline

(B)

-\frac{\log (x)}{\ln (10)}

\newline

(C)

-\frac{1}{x^{2} \ln (10)}

\newline

(D)

\log (x)

Differentiate $h(x)$ : Differentiate $h(x) = \log(x)$ with respect to $x$ to find the first derivative, $h'(x)$ . Using the derivative of the logarithm function, we have: $h'(x) = \frac{d}{dx} [\log(x)] = \frac{1}{x\ln(10)}$
Find first derivative: Differentiate $h'(x)$ to find the second derivative, $h''(x)$ . We need to apply the derivative to $h'(x) = \frac{1}{x\ln(10)}$ . This is a quotient, so we can use the quotient rule or recognize it as the derivative of a reciprocal function. $h''(x) = \frac{d}{dx} \left[\frac{1}{x\ln(10)}\right] = \frac{d}{dx} \left[\frac{x^{-1}}{\ln(10)}\right]$ Since $\ln(10)$ is a constant, we can differentiate $x^{-1}$ with respect to $x$ : $h''(x) = -1 \times x^{-2} / \ln(10)$
Find second derivative: Simplify the expression for $h''(x)$ . $h''(x) = -\frac{1}{x^2\ln(10)}$ This is the simplified form of the second derivative of $h(x)$ .

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