Question

finding a derivative in exercises the derivative of the function. 39. $f(x)=x^{2}+5 - 3x^{-2}$ 40. $f(x)=x^{3}-2x$

Explanation:

Step1: Apply sum - difference rule

The derivative of a sum/difference of functions is the sum/difference of their derivatives. So $f^\prime(x)=(x^{2})^\prime+(5)^\prime-(3x^{- 2})^\prime$.

Step2: Use power rule for $x^n$ derivative

The power rule states that $(x^{n})^\prime = nx^{n - 1}$. For $y = x^{2}$, $y^\prime=2x^{2 - 1}=2x$. For a constant $y = 5$, $y^\prime = 0$. For $y = 3x^{-2}$, $y^\prime=3\times(-2)x^{-2 - 1}=-6x^{-3}$.

Step3: Combine the results

$f^\prime(x)=2x+0 - (-6x^{-3})=2x + \frac{6}{x^{3}}$.

Answer:

$f^\prime(x)=2x+\frac{6}{x^{3}}$