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

Identify Functions: To find the derivative of the function h(x) = (cos(x))/(ln(x)), we will use the quotient rule, which states that the derivative of a function that is the quotient of two other functions, u(x)/v(x), is given by (v(x)u'(x) - u(x)v'(x))/(v(x))^2.Find Derivatives: First, we identify the functions u(x) and v(x) where u(x) = cos(x) and v(x) = ln(x). We will need to find the derivatives u'(x) and v'(x).Apply Quotient Rule: The derivative of u(x) = cos(x) with respect to x is u'(x) = -sin(x).Simplify Expression: The derivative of v(x) = ln(x) with respect to x is v'(x) = 1/x.Combine Terms: Now we apply the quotient rule: h'(x) = (v(x)u'(x) - u(x)v'(x))/(v(x))^2. Substituting the derivatives we found, we get h'(x) = (ln(x)(-sin(x)) - cos(x)(1/x))/(ln(x))^2.Match Answer Choices: Simplify the expression: h'(x) = (-ln(x)sin(x) - cos(x)/x)/(ln(x))^2.We can further simplify by combining the terms in the numerator over a common denominator, which is xln(x): h'(x) = (-xln(x)sin(x) - cos(x))/(x(ln(x))^2).Now we look at the answer choices to see which one matches our derivative: (A) (-x sin(x)ln(x)-cos(x))/(x(ln(x))^(2)) matches our result. (B) -x sin(x) does not match our result. (C) (ln(x)sin(x)-cos(x))/(x ln(x)) does not match our result. (D) sin(x)-(1)/(x) does not match our result.

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Math Problems

Calculus

Find derivatives of using multiple formulae

Full solution

Q. Let

h(x)=\frac{\cos (x)}{\ln (x)}

\newline

Find

h^{\prime}(x)

\newline

Choose

1

answer:

\newline

(A)

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

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(B)

-x \sin (x)

\newline

(C)

\frac{\ln (x) \sin (x)-\cos (x)}{x \ln (x)}

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(D)

\sin (x)-\frac{1}{x}

Identify Functions: To find the derivative of the function $h(x) = \frac{\cos(x)}{\ln(x)}$ , we will use the quotient rule, which states that the derivative of a function that is the quotient of two other functions, $\frac{u(x)}{v(x)}$ , is given by $\frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}$ .
Find Derivatives: First, we identify the functions $u(x)$ and $v(x)$ where $u(x) = \cos(x)$ and $v(x) = \ln(x)$ . We will need to find the derivatives $u'(x)$ and $v'(x)$ .
Apply Quotient Rule: The derivative of $u(x) = \cos(x)$ with respect to $x$ is $u'(x) = -\sin(x)$ .
Simplify Expression: The derivative of $v(x) = \ln(x)$ with respect to $x$ is $v'(x) = \frac{1}{x}$ .
Combine Terms: Now we apply the quotient rule: $h'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}$ . Substituting the derivatives we found, we get $h'(x) = \frac{\ln(x)(-\sin(x)) - \cos(x)(\frac{1}{x})}{(\ln(x))^2}$ .
Match Answer Choices: Simplify the expression: $h'(x) = \frac{-\ln(x)\sin(x) - \frac{\cos(x)}{x}}{(\ln(x))^2}$ .
Match Answer Choices: Simplify the expression: $h'(x) = \frac{-\ln(x)\sin(x) - \frac{\cos(x)}{x}}{(\ln(x))^2}$ .We can further simplify by combining the terms in the numerator over a common denominator, which is $x\ln(x)$ : $h'(x) = \frac{-x\ln(x)\sin(x) - \cos(x)}{x(\ln(x))^2}$ .
Match Answer Choices: Simplify the expression: $h'(x) = \frac{-\ln(x)\sin(x) - \cos(x)/x}{(\ln(x))^2}$ .We can further simplify by combining the terms in the numerator over a common denominator, which is $x\ln(x)$ : $h'(x) = \frac{-x\ln(x)\sin(x) - \cos(x)}{x(\ln(x))^2}$ .Now we look at the answer choices to see which one matches our derivative: $\newline$ (A) $\frac{-x \sin(x)\ln(x)-\cos(x)}{x(\ln(x))^{2}}$ matches our result. $\newline$ (B) $-x \sin(x)$ does not match our result. $\newline$ (C) $\frac{\ln(x)\sin(x)-\cos(x)}{x \ln(x)}$ does not match our result. $\newline$ (D) $\sin(x)-\frac{1}{x}$ does not match our result.