Find the slope of the curve at the given point P and an equation of the tangent line at P.
-$y = x ^ { 3 } - 9 x , P ( 1 , - 8 )$
A) slope is $- 6 ; y = - 6 x$
B) slope is $3 ; y = 3 x - 11$
C) slope is $3 ; y = 3 x - 7$
D) slope is $- 6 ; y = - 6 x - 2$

Question

Quizplus · Accepted Answer

To find the slope of the curve at point P, we differentiate $y = x^3 - 9x$ with respect to $x$, giving us $y' = 3x^2 - 9$. Substituting $x = 1$ into $y'$, we get $3(1)^2 - 9 = 3 - 9 = -6$, so the slope at P is $-6$. To find the equation of the tangent line at P, we use the point-slope form $y - y_1 = m(x - x_1)$, where $m = -6$ and $P(x_1, y_1) = (1, -8)$. Substituting these values, we get $y - (-8) = -6(x - 1)$, which simplifies to $y = -6x - 2$.

Find the Slope of the Curve at the Given Point y=x3−9x,P(1,−8)y = x ^ { 3 } - 9 x , P ( 1 , - 8 )y=x3−9x,P(1,−8)

Find the Slope of the Curve at the Given Point $y = x ^ { 3 } - 9 x , P ( 1 , - 8 )$