Problem: Find the equation of the tangent line to the curve $f(x) = \sin(x) + 3x \cos(x)$ at the point $(\pi, −3\pi)$.
The equation of this tangent line can be written in the form $y = mx + c$. Compute $m$ and $c$.
Attempt: To find the slope, differentiate: $f(x) = \sin(x) + 3x \cos(x)$
Apply the product rule only on products which is $$3x cos(x)$$
$$ m = f '(\pi) = \cos(x) + 3 cos(x) + 3x( -sin(x)) $$ $$ f '(\pi) = cos(\pi) + 3 cos (\pi) + 3(\pi)(-sin(\pi)) $$ $$ = -1 + 3 (-1) + 9.4247(0) $$ $$ = -1 -3 + 0 $$ $$ m = -4 $$
Then to compute m and c $$ y = mx + c $$ $$ -3\pi = -4 (\pi) + c $$ $$ -3\pi = -4\pi + c $$ $$ 4\pi -3\pi = c $$ $$ \pi = c $$
The equation of the tangent line is $$y = -4x + \pi$$
How am I doing?
EDITED
We can calculate $f'(x)=\cos(x)+3\cos(x)-3x \sin(x)$. Then $f'(\pi)=-4$. Therefore equation of the tangent line is $y=-4x+\pi$.