Let y=ln(sin(x)). Find (dy)/(dx). Choose 1 answer: (A) (cos(x))/(sin(x)) (B) (1)/(sin(x)) (C) (1)/(cos(x)) (D) ln(cos(x))

Identify Function: Identify the function and its composition. We have y = ln(sin(x)), which is a composition of two functions: the natural logarithm function and the sine function. We need to find the derivative of this composite function.Apply Chain Rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is ln(u) (where u = sin(x)) and the inner function is sin(x).Differentiate Outer Function: Differentiate the outer function with respect to the inner function. The derivative of ln(u) with respect to u is 1/u. So, when we apply this to our function, we get the derivative of ln(sin(x)) with respect to sin(x) as 1/sin(x).Differentiate Inner Function: Differentiate the inner function with respect to x. The derivative of sin(x) with respect to x is cos(x). So, we will multiply the result from Step 3 by cos(x).Combine Results: Combine the results from Steps 3 and 4. Multiplying 1/sin(x) by cos(x), we get (cos(x))/(sin(x)), which can also be written as cot(x).Match Final Result: Match the final result with the given options. The derivative (dy)/(dx) is (cos(x))/(sin(x)), which matches option (A).

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Let
y=ln(sin(x)).
Find
(dy)/(dx).
Choose 1 answer:
(A)
(cos(x))/(sin(x))
(B)
(1)/(sin(x))
(C)
(1)/(cos(x))
(D)
ln(cos(x))

Let $y=\ln (\sin (x))$ . $\newline$ Find $\frac{d y}{d x}$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{\cos (x)}{\sin (x)}$ $\newline$ (B) $\frac{1}{\sin (x)}$ $\newline$ (C) $\frac{1}{\cos (x)}$ $\newline$ (D) $\ln (\cos (x))$

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Math Problems

Calculus

Find derivatives of logarithmic functions

Full solution

Q. Let

y=\ln (\sin (x))

\newline

Find

\frac{d y}{d x}

\newline

Choose

1

answer:

\newline

(A)

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

\newline

(B)

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

\newline

(C)

\frac{1}{\cos (x)}

\newline

(D)

\ln (\cos (x))

Identify Function: Identify the function and its composition. $\newline$ We have $y = \ln(\sin(x))$ , which is a composition of two functions: the natural logarithm function and the sine function. We need to find the derivative of this composite function.
Apply Chain Rule: Apply the chain rule for differentiation. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is $\ln(u)$ (where $u = \sin(x)$ ) and the inner function is $\sin(x)$ .
Differentiate Outer Function: Differentiate the outer function with respect to the inner function. $\newline$ The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$ . So, when we apply this to our function, we get the derivative of $\ln(\sin(x))$ with respect to $\sin(x)$ as $\frac{1}{\sin(x)}$ .
Differentiate Inner Function: Differentiate the inner function with respect to $x$ . The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$ . So, we will multiply the result from Step $3$ by $\cos(x)$ .
Combine Results: Combine the results from Steps $3$ and $4$ . $\newline$ Multiplying $\frac{1}{\sin(x)}$ by $\cos(x)$ , we get $\frac{\cos(x)}{\sin(x)}$ , which can also be written as $\cot(x)$ .
Match Final Result: Match the final result with the given options. $\newline$ The derivative $\frac{dy}{dx}$ is $\frac{\cos(x)}{\sin(x)}$ , which matches option (A).