Question
Differentiate the function \( G(x)=6 x^{-4}+5 x^{-5} \)
Ask by Wilson Peterson. in the United States
Feb 22,2025
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Tutor-Verified Answer
Answer
The derivative of \( G(x) = 6x^{-4} + 5x^{-5} \) is \( G'(x) = -24x^{-5} - 25x^{-6} \) or \( G'(x) = -\frac{24}{x^5} - \frac{25}{x^6} \).
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The Deep Dive
To differentiate the function \( G(x) = 6x^{-4} + 5x^{-5} \), we apply the power rule of differentiation. The derivative of \( x^n \) is \( n x^{n-1} \). Thus, for the first term, \( G'(x) = 6 \cdot (-4)x^{-5} = -24x^{-5} \), and for the second term, \( G'(x) = 5 \cdot (-5)x^{-6} = -25x^{-6} \). Combining these, we have: \[ G'(x) = -24x^{-5} - 25x^{-6}. \] Now, don't forget to simplify this if needed; you can factor out common terms! For example, we could factor out \( -x^{-6} \) to get: \[ G'(x) = -x^{-6}(24x + 25). \]
