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

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}})