Question
Find the derivative of the following function. \[ g(x)=3 x^{3}+\frac{5}{x^{7}} \]
Ask by Cole Griffin. in the United States
Feb 21,2025
Upstudy AI Solution
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Answer
The derivative of \( g(x) = 3x^3 + \frac{5}{x^7} \) is \( 9x^2 - \frac{35}{x^8} \).
Solution
The derivative of the function \( g(x) = 3x^3 + \frac{5}{x^7} \) with respect to \( x \) cannot be calculated directly using the provided function.
However, we can calculate the derivative manually by applying the power rule for differentiation.
The derivative of \( 3x^3 \) is \( 9x^2 \) and the derivative of \( \frac{5}{x^7} \) is \( -\frac{35}{x^8} \).
Therefore, the derivative of the function \( g(x) \) is \( 9x^2 - \frac{35}{x^8} \).
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Simplify this solution The Deep Dive
To find the derivative of the function \( g(x) = 3x^3 + \frac{5}{x^7} \), we will apply the power rule. First, let's rewrite the term \( \frac{5}{x^7} \) as \( 5x^{-7} \). Now, we differentiate each term separately: - The derivative of \( 3x^3 \) is \( 3 \cdot 3x^{3-1} = 9x^2 \). - The derivative of \( 5x^{-7} \) is \( 5 \cdot (-7)x^{-7-1} = -35x^{-8} \). Combining these results, we have: \[ g'(x) = 9x^2 - 35x^{-8} \] Thus, the derivative of the function is: \[ g'(x) = 9x^2 - \frac{35}{x^8} \]
