Question
Find the derivatives of y=2x+1\x²
Answers
GPT-4o mini
To find the derivative of the function \( y = 2x + \frac{1}{x^2} \), we will differentiate each term separately.
1. The derivative of \( 2x \) is:
\[
\frac{d}{dx}(2x) = 2
\]
2. The derivative of \( \frac{1}{x^2} \) can be found using the power rule. First, rewrite \( \frac{1}{x^2} \) as \( x^{-2} \):
\[
\frac{d}{dx}(x^{-2}) = -2x^{-3} = -\frac{2}{x^3}
\]
Now, combine the derivatives of both terms:
\[
\frac{dy}{dx} = 2 - \frac{2}{x^3}
\]
Thus, the derivative of \( y = 2x + \frac{1}{x^2} \) is:
\[
\frac{dy}{dx} = 2 - \frac{2}{x^3}
\]
1. The derivative of \( 2x \) is:
\[
\frac{d}{dx}(2x) = 2
\]
2. The derivative of \( \frac{1}{x^2} \) can be found using the power rule. First, rewrite \( \frac{1}{x^2} \) as \( x^{-2} \):
\[
\frac{d}{dx}(x^{-2}) = -2x^{-3} = -\frac{2}{x^3}
\]
Now, combine the derivatives of both terms:
\[
\frac{dy}{dx} = 2 - \frac{2}{x^3}
\]
Thus, the derivative of \( y = 2x + \frac{1}{x^2} \) is:
\[
\frac{dy}{dx} = 2 - \frac{2}{x^3}
\]