Question

find the derivative of f(x) = \frac{1}{x} at x = \frac{6}{7}.
f\left(\frac{6}{7}\
ight)= (type an integer or a simplified fraction.)

Explanation:

Step1: Rewrite the function

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

Step2: Apply the power - rule

The power - rule for differentiation is $\frac{d}{dx}(x^n)=nx^{n - 1}$. For $n=-1$, we have $f^\prime(x)=-1\times x^{-1 - 1}=-x^{-2}=-\frac{1}{x^{2}}$.

Step3: Evaluate the derivative at $x = \frac{6}{7}$

Substitute $x=\frac{6}{7}$ into $f^\prime(x)$. So $f^\prime(\frac{6}{7})=-\frac{1}{(\frac{6}{7})^{2}}$.
Since $(\frac{6}{7})^{2}=\frac{36}{49}$, then $f^\prime(\frac{6}{7})=-\frac{49}{36}$.

Answer:

$-\frac{49}{36}$