the slope of the graph of the function $f(x)=sqrt{10x}$ at $(10,10)$ is $square$. (type an integer or a…

Explanation:

Step1: Differentiate the function

We have (f(x)=\sqrt{10x}=(10x)^{\frac{1}{2}}). Using the power rule ((u^n)^\prime = nu^{n - 1}u^\prime), where (u = 10x), (n=\frac{1}{2}), and (u^\prime=10). (f^\prime(x)=\frac{1}{2}(10x)^{-\frac{1}{2}}\times10=\frac{10}{2\sqrt{10x}}=\frac{5}{\sqrt{10x}}).

Step2: Evaluate the derivative at (x = 10)

Substitute (x = 10) into (f^\prime(x)). (f^\prime(10)=\frac{5}{\sqrt{10\times10}}=\frac{5}{10}=\frac{1}{2}).

Answer:

(\frac{1}{2})