Norris Guerrero
06/27/2023 · Middle School
Problem 2. (0 points) Differentiate \( y=\sqrt[4]{\cos (8 \pi x)} \). Use 'pi' for \( \pi \) Answer: \( \frac{d y}{d x}=\square \)
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Tutor-Verified Answer
Step-by-step Solution
To differentiate \( y = \sqrt[4]{\cos(8 \pi x)} \), we can rewrite it as:
\[
y = (\cos(8 \pi x))^{1/4}
\]
Now, we will use the chain rule to differentiate this function. The chain rule states that if \( y = f(g(x)) \), then:
\[
\frac{dy}{dx} = f'(g(x)) \cdot g'(x)
\]
In our case, let \( f(u) = u^{1/4} \) and \( g(x) = \cos(8 \pi x) \).
First, we find the derivative of \( f(u) \):
\[
f'(u) = \frac{1}{4} u^{-3/4}
\]
Next, we find the derivative of \( g(x) \):
\[
g'(x) = -\sin(8 \pi x) \cdot (8 \pi) = -8 \pi \sin(8 \pi x)
\]
Now, applying the chain rule:
\[
\frac{dy}{dx} = f'(g(x)) \cdot g'(x) = \frac{1}{4} (\cos(8 \pi x))^{-3/4} \cdot (-8 \pi \sin(8 \pi x))
\]
Simplifying this expression:
\[
\frac{dy}{dx} = -2 \pi (\cos(8 \pi x))^{-3/4} \sin(8 \pi x)
\]
Thus, the final answer is:
\[
\frac{dy}{dx} = -2 \pi \frac{\sin(8 \pi x)}{(\cos(8 \pi x))^{3/4}}
\]
So, the answer is:
\[
\frac{d y}{d x} = -2 \pi \frac{\sin(8 \pi x)}{(\cos(8 \pi x))^{3/4}}
\]
Quick Answer
\(\frac{d y}{d x} = -2 \pi \frac{\sin(8 \pi x)}{(\cos(8 \pi x))^{3/4}}\)
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