Find the second derivative of the function. $f(x)=5 e^{x} \cos x$
A) $10 e^{x} \sin x$
B) $10 e^{x} \cos x$
C) $-10 e^{x} \cos x$
D) $-10 e^{x} \sin x$

Question

Quizplus · Accepted Answer

First, we find the first derivative of the function:
$$f'(x) = 5 e^x \cos x - 5 e^x \sin x$$
Next, we find the second derivative:
$$f''(x) = (5 e^x \cos x - 5 e^x \sin x)' = (5 e^x \cos x)' - (5 e^x \sin x)'$$
Using the product rule, we get:
$$(5 e^x \cos x)' = 5 e^x \cos x - 5 e^x \sin x$$
$$(5 e^x \sin x)' = 5 e^x \sin x + 5 e^x \cos x$$
Therefore, we have:
$$f''(x) = (5 e^x \cos x - 5 e^x \sin x)' = (5 e^x \cos x)' - (5 e^x \sin x)' = 5 e^x \cos x - 5 e^x \sin x - 5 e^x \sin x - 5 e^x \cos x = -10 e^x \sin x$$
Thus, the answer is D: $-10 e^{x} \sin x$.

Find the Second Derivative of the Function f(x)=5excos⁡xf(x)=5 e^{x} \cos xf(x)=5excosx

Find the Second Derivative of the Function $f(x)=5 e^{x} \cos x$