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

Apply Quotient Rule: Apply the quotient rule to find the derivative of g(x) = (sin(x))/(e^(x)). The quotient rule states that if you have a function h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x))/(g(x))^2.Identify Functions: Identify the functions f(x) and g(x) where f(x) = sin(x) and g(x) = e^(x). We need to find the derivatives f'(x) and g'(x).Find f'(x): Find the derivative of f(x) = sin(x). The derivative of sin(x) with respect to x is cos(x), so f'(x) = cos(x).Find g'(x): Find the derivative of g(x) = e^(x). The derivative of e^(x) with respect to x is e^(x), so g'(x) = e^(x).Apply Quotient Rule: Apply the quotient rule using the derivatives from steps 3 and 4. g'(x) = (f'(x)g(x) - f(x)g'(x))/(g(x))^2 = (cos(x)e^(x) - sin(x)e^(x))/(e^(x))^2.Simplify Expression: Simplify the expression from step 5. Since e^(x) is common in both terms in the numerator, we can factor it out. g'(x) = e^(x)(cos(x) - sin(x))/(e^(x))^2.Match Correct Answer: Simplify further by canceling out one e^(x) from the numerator and denominator. g'(x) = (cos(x) - sin(x))/(e^(x)).Match the simplified derivative to the given answer choices. The correct answer is (D) (cos(x) - sin(x))/(e^(x)).

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

Let $g(x)=\frac{\sin (x)}{e^{x}}$ . $\newline$ Find $g^{\prime}(x)$ . $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{\cos (x)}{e^{x}}$ $\newline$ (B) $\cos (x)-e^{x}$ $\newline$ (C) $e^{x}(\cos (x)-\sin (x))$ $\newline$ (D) $\frac{\cos (x)-\sin (x)}{e^{x}}$

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Math Problems

Calculus

Find derivatives of using multiple formulae

Full solution

Q. Let

g(x)=\frac{\sin (x)}{e^{x}}

\newline

Find

g^{\prime}(x)

\newline

Choose

1

answer:

\newline

(A)

\frac{\cos (x)}{e^{x}}

\newline

(B)

\cos (x)-e^{x}

\newline

(C)

e^{x}(\cos (x)-\sin (x))

\newline

(D)

\frac{\cos (x)-\sin (x)}{e^{x}}

Apply Quotient Rule: Apply the quotient rule to find the derivative of $g(x) = \frac{\sin(x)}{e^{x}}$ . The quotient rule states that if you have a function $h(x) = \frac{f(x)}{g(x)}$ , then $h'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ .
Identify Functions: Identify the functions $f(x)$ and $g(x)$ where $f(x) = \sin(x)$ and $g(x) = e^{x}$ . We need to find the derivatives $f'(x)$ and $g'(x)$ .
Find $f'(x)$ : Find the derivative of $f(x) = \sin(x)$ . The derivative of $\sin(x)$ with respect to $x$ is $\cos(x)$ , so $f'(x) = \cos(x)$ .
Find $g'(x)$ : Find the derivative of $g(x) = e^{x}$ . The derivative of $e^{x}$ with respect to $x$ is $e^{x}$ , so $g'(x) = e^{x}$ .
Apply Quotient Rule: Apply the quotient rule using the derivatives from steps $3$ and $4$ . $g'(x) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2} = \frac{\cos(x)e^{x} - \sin(x)e^{x}}{(e^{x})^2}$ .
Simplify Expression: Simplify the expression from step $5$ . Since $e^{x}$ is common in both terms in the numerator, we can factor it out. $g'(x) = \frac{e^{x}(\cos(x) - \sin(x))}{(e^{x})^2}.$
Match Correct Answer: Simplify further by canceling out one $e^{x}$ from the numerator and denominator. $g'(x) = \frac{\cos(x) - \sin(x)}{e^{x}}$ .
Match Correct Answer: Simplify further by canceling out one $e^{x}$ from the numerator and denominator. $g'(x) = \frac{\cos(x) - \sin(x)}{e^{x}}$ .Match the simplified derivative to the given answer choices. The correct answer is (D) $\frac{\cos(x) - \sin(x)}{e^{x}}$ .