Question

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

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:

\(\frac{1}{2}\)