Question
Consider the following \[ f(x)=x^{5}-5 x^{3}+x-1 \] Find \( f^{\prime}(x) \) \( f^{\prime}(x)=\square \)
Ask by Tran Wade. in the United States
Mar 09,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( f^{\prime}(x) = 5x^4 - 15x^2 + 1 \)
Solution
1. Start with the function:
\[
f(x)=x^{5}-5x^{3}+x-1
\]
2. Differentiate each term separately:
- The derivative of \( x^5 \) is \( 5x^4 \).
- The derivative of \( -5x^3 \) is \( -5 \cdot 3x^2 = -15x^2 \).
- The derivative of \( x \) is \( 1 \).
- The derivative of the constant \( -1 \) is \( 0 \).
3. Combine the derivatives:
\[
f^{\prime}(x)=5x^4-15x^2+1
\]
Thus,
\[
f^{\prime}(x)=\boxed{5x^4-15x^2+1}
\]
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Simplify this solution Bonus Knowledge
To find the derivative of the function \( f(x) = x^5 - 5x^3 + x - 1 \), we apply the power rule to each term. The derivative of \( x^n \) is \( nx^{n-1} \). Thus, we calculate: 1. The derivative of \( x^5 \) is \( 5x^4 \). 2. The derivative of \( -5x^3 \) is \( -15x^2 \). 3. The derivative of \( x \) is \( 1 \). 4. The derivative of the constant \( -1 \) is \( 0 \). Combining these results, we get: \[ f^{\prime}(x) = 5x^4 - 15x^2 + 1. \] So, \( f^{\prime}(x)=5x^4 - 15x^2 + 1 \).
