find the derivative of $f(x)=8sqrt{x}-\frac{6}{x^{10}}$

Explanation:

Step1: Rewrite the function

Rewrite $f(x)$ as $f(x)=8x^{\frac{1}{2}}-6x^{- 10}$.

Step2: Apply the power - rule

The power - rule for differentiation is $\frac{d}{dx}(x^n)=nx^{n - 1}$. For the first term $8x^{\frac{1}{2}}$, its derivative is $8\times\frac{1}{2}x^{\frac{1}{2}-1}=4x^{-\frac{1}{2}}$. For the second term $-6x^{-10}$, its derivative is $-6\times(-10)x^{-10 - 1}=60x^{-11}$.

Step3: Combine the derivatives

The derivative of $f(x)$ is $f^\prime(x)=4x^{-\frac{1}{2}}+60x^{-11}$.

Answer:

$f^\prime(x)=4x^{-\frac{1}{2}}+60x^{-11}$