a. find the derivative of f at x. that is, find f(x). b. find the slope of the tangent line to the graph of…

a. find the derivative of f at x. that is, find f(x). b. find the slope of the tangent line to the graph of f at each of the two values of x given to the right of the function f(x)=√x; x = 1, x = 64 a. f(x)=

a. find the derivative of f at x. that is, find f(x). b. find the slope of the tangent line to the graph of f at each of the two values of x given to the right of the function f(x)=√x; x = 1, x = 64 a. f(x)=

Answer

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\sqrt{x}$ as $f(x)=x^{\frac{1}{2}}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = x^n$, then $y^\prime=nx^{n - 1}$. For $f(x)=x^{\frac{1}{2}}$, we have $f^\prime(x)=\frac{1}{2}x^{\frac{1}{2}-1}=\frac{1}{2}x^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}$.

Step3: Find the slope at $x = 1$

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

Step4: Find the slope at $x = 64$

Substitute $x = 64$ into $f^\prime(x)$. $f^\prime(64)=\frac{1}{2\sqrt{64}}=\frac{1}{2\times8}=\frac{1}{16}$.

Answer:

a. $f^\prime(x)=\frac{1}{2\sqrt{x}}$ b. At $x = 1$, the slope is $\frac{1}{2}$; at $x = 64$, the slope is $\frac{1}{16}$