Derivatives

g=a Cos[θ]

f=x^2

D[f,x]

g=1/x Sin[x]

dgdx=D[g,x]

dgdx/. {x->3,a->5}

N[%]

Plot[{g,dgdx},{x,0,10},PlotLegends->{"g","dgdx"}]

f2

f2=Sin[x] Sin[y]

Partial[f2,x]

(∂_x) f2

Plot3D[f2,{x,-Pi,Pi},{y,-Pi,Pi}]

Solving Equations

eqn=3x^3-2x^2-9==0

Solve[eqn,x]

N[%]

eqn1=3x^3-2x y +6 y==0

eqn2=-10x y^2 -3 x ==0

Solve[{eqn1,eqn2},{x,y}]

N[%]

Integrals

f=x^2

∫f\[DifferentialD]x

(∫_0^10)f\[DifferentialD]x

Integrate[f,x]

((x^3)/3)

Integrate[f,{x,0,10}]

Vectors

a={1,2,3}

b={-3,1,2}

orig={0,0,0}

Norm[a]

Normalize[a]

a.b

c=Cross[a,b]

ab=Projection[a,b]

Normalize[b]==Normalize[ab]

Show[Graphics3D[{
{Blue,Line[{orig,a}]},
{Red,Line[{orig,b}]},
{Black,Thick,Line[{orig,ab}]}
}]]

d=Cross[a,Cross[c,a]]

Show[Graphics3D[{
{Blue,Line[{orig,a}]},
{Red,Line[{orig,b}]},
{Green,Line[{orig,c}]}
}]]

Plotting

f=-1/r^2

w=Integrate[f,r]

Plot[f,{r,-5,5}]

Plot[f,{r,1,3},Filling->Axis,PlotTheme->Scientific,PlotRange->{{0,5},{0,-1.5}}]