Let \( y+\cos x y^{2}+3 x^{2}=4 \) \( \operatorname{then} \frac{d y}{d x} \) \( \frac{\frac{y^{2} \sin x y^{2}-6 x}{1-2 x y \sin x y^{2}}}{1-2 x y \sin x y^{2}} \) \( \frac{2 y \sin x y^{2}+6 x}{1-2 x y \sin x y^{2}} \)

Let \( f(x)=2 x^{3}+3 x^{2}-12 x+1 \). a. Find all points on the graph of \( f(x) \) at which the tangent line is horizontal. b. Find all points on the graph of \( f(x) \) at which the tangent line has slope 108 .

Find the intervals on which \( f \) is increasing and the intervals on which it is decreasing. \( f(x)=\frac{x^{4}}{4}-3 x^{3}+9 x^{2} \)

Given \( f(x, y)=3 x^{3}+x y^{2}-2 y^{6} \), find \( f_{y} \) \( 2 x-60 y \) \( 18 x \) \( 2 y \) None of these answers

Find an equation of the line tangent to the given curve at \( \mathrm{x}=\mathrm{a} \). \[ y=4+6 x+11 x e^{x} ; a=0 \] The equation for the tangent line is \( y=\square \).

Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. \( f(x)=x^{4}-3 x^{3}+9 \)