Question
For the function \( f(x)=x^{4}+\frac{2}{x} \) Find a) \( f \prime(x) \) b) \( f \prime \prime(x) \)
Ask by French Burns. in the United States
Feb 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
a) \( f'(x) = 4x³ - \frac{2}{x²} \)
b) \( f''(x) = 12x² + \frac{4}{x³} \)
Solution
We start with the function
f(x) = x⁴ + (2/x).
It is often easier to rewrite f(x) using exponents:
f(x) = x⁴ + 2x⁻¹.
a) To find f '(x), differentiate term by term:
• The derivative of x⁴ is 4x³.
• The derivative of 2x⁻¹ is 2(–1)x^(–2) = –2x⁻².
Thus,
f '(x) = 4x³ – 2x⁻²,
or equivalently,
f '(x) = 4x³ – 2/x².
b) To find f ''(x), we differentiate f '(x):
f '(x) = 4x³ – 2x⁻².
Differentiate each term:
• The derivative of 4x³ is 12x².
• The derivative of –2x⁻² is –2(–2)x^(–3) = 4x⁻³.
Thus,
f ''(x) = 12x² + 4x⁻³,
or equivalently,
f ''(x) = 12x² + 4/x³.
So, the answers are:
a) f '(x) = 4x³ – 2/x²
b) f ''(x) = 12x² + 4/x³.
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Simplify this solution Bonus Knowledge
To find the derivatives of the function \( f(x) = x^{4} + \frac{2}{x} \): a) First, we can use the power rule and the quotient rule to find the first derivative \( f'(x) \): \[ f'(x) = \frac{d}{dx}(x^{4}) + \frac{d}{dx}\left(\frac{2}{x}\right) = 4x^{3} - \frac{2}{x^{2}} \] Thus, \[ f'(x) = 4x^{3} - 2x^{-2} = 4x^{3} - \frac{2}{x^{2}}. \] b) Next, we find the second derivative \( f''(x) \): \[ f''(x) = \frac{d}{dx}(4x^{3}) + \frac{d}{dx}\left(-\frac{2}{x^{2}}\right) = 12x^{2} + \frac{4}{x^{3}}. \] Combining these, we get: \[ f''(x) = 12x^{2} + 4x^{-3} = 12x^{2} + \frac{4}{x^{3}}. \]
