Find the slope of the curve at the indicated point.
-Find equations of all tangents to the curve $f ( x ) = \frac { 1 } { x + 1 }$ that have slope $- 1$.
A) $y = - x + 3$
B) $y = - x + 1 , y = - x - 3$
C) $y = - x + 3 , y = - x + 1$
D) $y = - x + 1$

Question

Quizplus · Accepted Answer

The derivative of $f(x) = \frac{1}{x+1}$ is $f'(x) = -\frac{1}{(x+1)^2}$. At any point $(a, f(a))$ on the curve, the slope of the tangent line is $f'(a) = -\frac{1}{(a+1)^2}$.To find the equations of all tangents to the curve that have slope $-1$, we set $f'(a) = -1$ and solve for $a$:$$-\frac{1}{(a+1)^2} = -1$$$$(a+1)^2 = 1$$$$a+1 = \pm 1$$$$a = 0, -2$$So the tangents to the curve that have slope $-1$ are the lines with equations$$y - f(0) = -1(x-0) \quad \Rightarrow \quad y = -x + 3$$and$$y - f(-2) = -1(x-(-2)) \quad \Rightarrow \quad y = -x + 1$$

Find the Slope of the Curve at the Indicated Point f(x)=1x+1f ( x ) = \frac { 1 } { x + 1 }f(x)=x+11​

Find the Slope of the Curve at the Indicated Point $f ( x ) = \frac { 1 } { x + 1 }$