Find the derivative of the function.

[tex]\[
\begin{array}{l}
y=\frac{4}{x^5}-\frac{6}{x} \\
\frac{d y}{d x} = \square
\end{array}
\][/tex]



Answer :

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]