To find the derivative of the function [tex]\( y = 11x^2 - 6x - 7x^{-2} \)[/tex], we will apply the rules of differentiation to each term separately. Here are the steps:
1. Identify the function and its terms:
[tex]\[
y = 11x^2 - 6x - 7x^{-2}
\][/tex]
2. Differentiate each term separately using the power rule:
The power rule states that the derivative of [tex]\(x^n\)[/tex] with respect to [tex]\(x\)[/tex] is [tex]\(nx^{n-1}\)[/tex].
- For the first term [tex]\(11x^2\)[/tex]:
[tex]\[
\frac{d}{dx}(11x^2) = 11 \cdot 2x^{2-1} = 22x
\][/tex]
- For the second term [tex]\(-6x\)[/tex]:
[tex]\[
\frac{d}{dx}(-6x) = -6 \cdot 1x^{1-1} = -6
\][/tex]
- For the third term [tex]\(-7x^{-2}\)[/tex]:
[tex]\[
\frac{d}{dx}(-7x^{-2}) = -7 \cdot (-2)x^{-2-1} = 14x^{-3}
\][/tex]
3. Combine the derivatives of each term:
[tex]\[
\frac{dy}{dx} = 22x - 6 + 14x^{-3}
\][/tex]
Hence, the derivative of the function [tex]\( y = 11x^2 - 6x - 7x^{-2} \)[/tex] is:
[tex]\[
\frac{dy}{dx} = 22x - 6 + 14x^{-3}
\][/tex]
