Question

If f(x) = 8x2 − x3, find f'(2) and use it to find an equation of the tangent line to the curve y = 8x2 − x3 at the point (2, 24).

Answer 1

Take derivitive to find slope
$f'(x)=16x-3x^2$

at x=2
$f'(2)=16(2)-3(2)^2$
$f'(2)=32-3(4)$
$f'(2)=32-12$
$f'(2)=20$

so the slope of f(x) at the point (2,24) is 20

use point slope form
the equation of a line that passes through the point $(x_1,y_1)$ and has a slope of m is
$y-y_1=m(x-x_1)$
we are given that the slope is 20 and the point is (2,24)

the equation is
$y-24=20(x-2)$
simplify into different form yourself if needed