Certainly! Let's find the derivative of the function [tex]\( f(x) = 5x - \frac{1}{x^2} \)[/tex] step-by-step.
Step 1: Identify the function components\
The function given is:
[tex]\[ f(x) = 5x - \frac{1}{x^2} \][/tex]
Step 2: Differentiate each term separately
Differentiate [tex]\( 5x \)[/tex]:\
The derivative of [tex]\( 5x \)[/tex] with respect to [tex]\( x \)[/tex] is simply 5, because the derivative of [tex]\( x \)[/tex] with respect to [tex]\( x \)[/tex] is 1, and we multiply it by the constant coefficient.
Differentiate [tex]\( -\frac{1}{x^2} \)[/tex]:\
To differentiate [tex]\( -\frac{1}{x^2} \)[/tex], we can rewrite it using exponent notation. Recall that [tex]\( \frac{1}{x^2} = x^{-2} \)[/tex]. Hence,
[tex]\[ -\frac{1}{x^2} = -x^{-2} \][/tex]
Applying the power rule of differentiation [tex]\( \frac{d}{dx} [x^n] = nx^{n-1} \)[/tex]:
[tex]\[
\frac{d}{dx}(-x^{-2}) = -2x^{-3}
\][/tex]
Simplifying, we get:
[tex]\[
\frac{d}{dx}(-x^{-2}) = -2 \cdot \frac{1}{x^3} = \frac{2}{x^3}
\][/tex]
Step 3: Combine the results\
Adding the derivatives from each term, we have:
[tex]\[ \frac{d}{dx} \left( 5x - \frac{1}{x^2} \right) = 5 + \frac{2}{x^3} \][/tex]
So, the derivative of the function [tex]\( 5x - \frac{1}{x^2} \)[/tex] is:
[tex]\[ \boxed{5 + \frac{2}{x^3}} \][/tex]
This completes our step-by-step differentiation of the given function.
