To find the derivative of the function [tex]\( y = \frac{4}{x^5} - \frac{6}{x} \)[/tex], we will use the rules of differentiation step by step. Here’s how we can approach it:
1. Rewrite the Function: First, let's express the function using exponents to make differentiation easier:
[tex]\[
y = 4x^{-5} - 6x^{-1}
\][/tex]
2. Differentiate Term by Term: Use the power rule for differentiation, which states that [tex]\(\frac{d}{dx} (x^n) = nx^{n-1}\)[/tex].
- For the first term [tex]\(4x^{-5}\)[/tex]:
[tex]\[
\frac{d}{dx} (4x^{-5}) = 4 \cdot (-5)x^{-6} = -20x^{-6}
\][/tex]
- For the second term [tex]\((-6x^{-1})\)[/tex]:
[tex]\[
\frac{d}{dx} (-6x^{-1}) = -6 \cdot (-1)x^{-2} = 6x^{-2}
\][/tex]
3. Combine the Results: Combine the derivatives of each term:
[tex]\[
\frac{dy}{dx} = -20x^{-6} + 6x^{-2}
\][/tex]
4. Simplify the Expression: Finally, we can rewrite the expression in its simplified form, using exponents as required:
[tex]\[
\frac{dy}{dx} = \frac{6}{x^2} - \frac{20}{x^6}
\][/tex]
Therefore, the derivative of the function [tex]\( y = \frac{4}{x^5} - \frac{6}{x} \)[/tex] is:
[tex]\[
\frac{dy}{dx} = \frac{6}{x^2} - \frac{20}{x^6}
\][/tex]
