Question

find the derivative of $f(x)=-10\sqrt{x}+\frac{1}{x^{6}}$.
type your answer without fractional or negative exponents.
$f(x)=$

Explanation:

Step1: Rewrite the function

Rewrite \(f(x)=- 10\sqrt{x}+\frac{1}{x^{6}}\) using exponent rules. We know that \(\sqrt{x}=x^{\frac{1}{2}}\) and \(\frac{1}{x^{6}} = x^{-6}\). So \(f(x)=-10x^{\frac{1}{2}}+x^{-6}\).

Step2: Apply the power rule

The power rule states that if \(y = ax^{n}\), then \(y^\prime=anx^{n - 1}\).
For the term \(-10x^{\frac{1}{2}}\), using the power rule: \(a=-10\), \(n = \frac{1}{2}\). The derivative is \(-10\times\frac{1}{2}x^{\frac{1}{2}-1}=-5x^{-\frac{1}{2}}\).
For the term \(x^{-6}\), using the power rule: \(a = 1\), \(n=-6\). The derivative is \(1\times(-6)x^{-6 - 1}=-6x^{-7}\).

Step3: Rewrite without negative exponents

We know that \(x^{-\frac{1}{2}}=\frac{1}{\sqrt{x}}\) and \(x^{-7}=\frac{1}{x^{7}}\). But the problem asks for no fractional or negative exponents.
\(f^\prime(x)=-5x^{-\frac{1}{2}}-6x^{-7}\) can be rewritten as \(f^\prime(x)=-\frac{5}{\sqrt{x}}-\frac{6}{x^{7}}\)

Answer:

\(f^\prime(x)=-\frac{5}{\sqrt{x}}-\frac{6}{x^{7}}\)