Question

find the slope of the graph of $f(x)=-7sqrt{x}$ at the point $(64, - 56)$. give an exact answer. enter dne if the slope does not exist.

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=-7\sqrt{x}$ as $f(x)=-7x^{\frac{1}{2}}$.

Step2: Find the derivative

Use the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$. So $f^\prime(x)=-7\times\frac{1}{2}x^{\frac{1}{2}-1}=-\frac{7}{2}x^{-\frac{1}{2}}=-\frac{7}{2\sqrt{x}}$.

Step3: Evaluate the derivative at $x = 64$

Substitute $x = 64$ into $f^\prime(x)$. $f^\prime(64)=-\frac{7}{2\sqrt{64}}=-\frac{7}{2\times8}=-\frac{7}{16}$.

Answer:

$-\frac{7}{16}$