Griffin Tyler
07/25/2023 · Senior High School
Let \( f(x)=\cos (x) \), find each of the following: Find the second derivative, \( f^{\prime \prime}(x)= \) Find the third derivative, \( f^{\prime \prime \prime}(x)= \) Find the fourth derivative, \( f^{\prime \prime \prime}(x)= \)
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Step-by-step Solution
To find the derivatives of the function \( f(x) = \cos(x) \), we will compute the first, second, third, and fourth derivatives step by step.
1. **First Derivative**:
\[
f'(x) = \frac{d}{dx} \cos(x) = -\sin(x)
\]
2. **Second Derivative**:
\[
f''(x) = \frac{d}{dx} f'(x) = \frac{d}{dx} (-\sin(x)) = -\cos(x)
\]
3. **Third Derivative**:
\[
f'''(x) = \frac{d}{dx} f''(x) = \frac{d}{dx} (-\cos(x)) = \sin(x)
\]
4. **Fourth Derivative**:
\[
f^{(4)}(x) = \frac{d}{dx} f'''(x) = \frac{d}{dx} \sin(x) = \cos(x)
\]
Now, we can summarize the results:
- The second derivative is:
\[
f''(x) = -\cos(x)
\]
- The third derivative is:
\[
f'''(x) = \sin(x)
\]
- The fourth derivative is:
\[
f^{(4)}(x) = \cos(x)
\]
Thus, the derivatives are:
- \( f^{\prime \prime}(x) = -\cos(x) \)
- \( f^{\prime \prime \prime}(x) = \sin(x) \)
- \( f^{(4)}(x) = \cos(x) \)
Quick Answer
- \( f^{\prime \prime}(x) = -\cos(x) \)
- \( f^{\prime \prime \prime}(x) = \sin(x) \)
- \( f^{(4)}(x) = \cos(x) \)
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