Question

this is section 2.5 problem 44: for the function f(x)=x^2 - \frac{1}{x}: (a) find the equation of the tangent line at the point x = - 1: y=

Explanation:

Step1: Find the derivative of the function

The derivative of $f(x)=x^{2}-\frac{1}{x}=x^{2}-x^{-1}$ using the power - rule. The power - rule states that if $y = x^{n}$, then $y^\prime=nx^{n - 1}$. So $f^\prime(x)=2x+x^{-2}=2x+\frac{1}{x^{2}}$.

Step2: Evaluate the derivative at $x = - 1$

Substitute $x=-1$ into $f^\prime(x)$. $f^\prime(-1)=2\times(-1)+\frac{1}{(-1)^{2}}=-2 + 1=-1$. This is the slope $m$ of the tangent line.

Step3: Find the $y$ - value of the function at $x=-1$

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

Step4: Use the point - slope form of a line

The point - slope form of a line is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(-1,2)$ and $m=-1$. So $y - 2=-1(x+1)$.

Step5: Simplify the equation

Expand and simplify: $y-2=-x - 1$, then $y=-x+1$.

Answer:

$y=-x + 1$