Let's tackle the given problem step by step. We want to find the derivative and the integral of the function [tex]\( y = 4x^3 - \frac{5}{x^2} \)[/tex].
### Part (a): Finding the Derivative [tex]\(\frac{dy}{dx}\)[/tex]
First, let's differentiate [tex]\( y \)[/tex] with respect to [tex]\( x \)[/tex].
The function given is:
[tex]\[ y = 4x^3 - \frac{5}{x^2} \][/tex]
To differentiate [tex]\( y \)[/tex], we can use the power rule for each term:
- For [tex]\(4x^3\)[/tex], the derivative is [tex]\( 12x^2 \)[/tex].
- For [tex]\(-\frac{5}{x^2}\)[/tex], we can rewrite [tex]\(\frac{5}{x^2}\)[/tex] as [tex]\(5x^{-2}\)[/tex]. The derivative of [tex]\(5x^{-2}\)[/tex] is [tex]\(-10x^{-3}\)[/tex], which simplifies to [tex]\(-\frac{10}{x^3}\)[/tex].
Combining these results, we get:
[tex]\[ \frac{dy}{dx} = 12x^2 + \frac{10}{x^3} \][/tex]
Thus, the derivative is:
[tex]\[ \boxed{12x^2 + \frac{10}{x^3}} \][/tex]
### Part (b): Finding the Integral [tex]\(\int y \, dx\)[/tex]
Next, we integrate the function [tex]\( y = 4x^3 - \frac{5}{x^2} \)[/tex] with respect to [tex]\( x \)[/tex].
We perform the integration term by term:
- For [tex]\(4x^3\)[/tex], the integral is [tex]\(\frac{4}{4}x^4 = x^4\)[/tex].
- For [tex]\(-\frac{5}{x^2}\)[/tex], we rewrite it as [tex]\(-5x^{-2}\)[/tex] and then integrate. The integral of [tex]\(x^{-2}\)[/tex] is [tex]\(-x^{-1} = -\frac{1}{x}\)[/tex], so the integral of [tex]\(-5x^{-2}\)[/tex] is [tex]\(\frac{5}{x}\)[/tex].
Combining these results along with the constant of integration [tex]\( C \)[/tex], we get:
[tex]\[ \int y \, dx = x^4 + \frac{5}{x} + C \][/tex]
Thus, the integral is:
[tex]\[ \boxed{x^4 + \frac{5}{x} + C} \][/tex]
This completes the solution for both parts (a) and (b).
