To find the derivative of the function [tex]\( f(x) = -2 x^{-8} - 3 x^{-5} \)[/tex], we'll use the power rule of differentiation. The power rule states that if [tex]\( f(x) = x^n \)[/tex], then [tex]\( f'(x) = nx^{n-1} \)[/tex].
Let's apply this rule to each term of the function.
1. Differentiate the first term [tex]\(-2 x^{-8}\)[/tex]:
Using the power rule:
[tex]\[
\frac{d}{dx}\left(-2 x^{-8}\right) = -2 \cdot (-8) x^{-8-1} = 16 x^{-9}
\][/tex]
2. Differentiate the second term [tex]\(-3 x^{-5}\)[/tex]:
Using the power rule:
[tex]\[
\frac{d}{dx}\left(-3 x^{-5}\right) = -3 \cdot (-5) x^{-5-1} = 15 x^{-6}
\][/tex]
3. Combine the derivatives:
Therefore, the derivative of the function [tex]\( f(x) \)[/tex] is:
[tex]\[
f'(x) = 16 x^{-9} + 15 x^{-6}
\][/tex]
To express the result more clearly, we can write the terms with positive exponents by moving them to the denominator:
[tex]\[
f'(x) = \frac{15}{x^6} + \frac{16}{x^9}
\][/tex]
Thus, the derivative of [tex]\( f(x) \)[/tex] is:
[tex]\[
f'(x) = \frac{15}{x^6} + \frac{16}{x^9}
\][/tex]
