Question
Compute \( \frac{d^{2}}{d x^{2}}\left(-x^{6}-3 x-\sqrt{2}\right) \)
Ask by Wood Ingram. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The second derivative of the function \( f(x) = -x^{6} - 3x - \sqrt{2} \) is \( f''(x) = -30x^{4} \).
Solution
We are given the function f(x) = -x⁶ - 3x - √2.
Step 1: Compute the first derivative f′(x).
f′(x) = d/dx ( -x⁶ ) + d/dx ( -3x ) + d/dx ( -√2 )
= -6x⁵ - 3 + 0
= -6x⁵ - 3
Step 2: Compute the second derivative f″(x).
f″(x) = d/dx ( -6x⁵ - 3 )
= -6 * 5x⁴ + 0
= -30x⁴
So, the second derivative of the given function is f″(x) = -30x⁴.
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Bonus Knowledge
To compute the second derivative of the function \( f(x) = -x^6 - 3x - \sqrt{2} \), we first find the first derivative. 1. **First Derivative**: \[ f'(x) = \frac{d}{dx}(-x^6) + \frac{d}{dx}(-3x) + \frac{d}{dx}(-\sqrt{2}) \] Using the power rule and the derivative of a constant: \[ f'(x) = -6x^5 - 3 + 0 = -6x^5 - 3 \] 2. **Second Derivative**: Now, we take the derivative of \( f'(x) \): \[ f''(x) = \frac{d}{dx}(-6x^5) + \frac{d}{dx}(-3) \] Again applying the power rule: \[ f''(x) = -30x^4 + 0 = -30x^4 \] Therefore, the second derivative is: \[ f''(x) = -30x^4 \]
