D[Sin[x], x]

Sin'[x]

D[x^3 + x, x]

(x^3 + x)'

D[Sin[x], {x, 1003}]

Derivative[1][Sin]

Derivative[1][Sin][x]

D[Sin[x*y], y]

D[Sin[x*y], {x, 3}, {y, 5}]

Integrate[x Cos[x y], y]

Integrate[x Cos[x y], x]

Integrate[1/Log[x], x]

Integrate[Exp[-x^2], {x, 0, Infinity}]

Series[Log[x], {x, 1, 10}]

TrigExpand[Sin[x + y + z]]

TrigReduce[
      Cos[y] Cos[z] Sin[x] + Cos[x] Cos[z] Sin[y] + 
        Cos[x] Cos[y] Sin[z] - Sin[x] Sin[y] Sin[z]]

Integrate[Cos[x]^10, x]

TrigExpand[%]

ReplaceAll[%, Sin[x]^2 -> 1 - Cos[x]^2]

ReplaceAll[%22, {Sin[x] -> s, Cos[x] -> c}]

ReplaceAll[%, s^2 -> 1 - c^2]

Collect[%, s]

ReplaceAll[%, s -> Sqrt[1 - c^2]]

ReplaceAll[
          Factor
[((((105 c)/256) + ((15 (c^3))/64) + ((45 (c^5))/512) + ((5 (c^7))/256) + 
                        ((c^9)/512)) s + (-((15 c)/64) - ((75 (c^3))/256) - ((35 (c^5))/256) - ((3 (c^7))/128)) (s^3) + (((45 c)/512) + ((35 (c^3))/256) + ((63 (c^5))/1280)) (s^5) + (-((5 c)/256) - ((3 (c^3))/128)) (s^7) + ((c (s^9))/512))/s], 
          s -> Sqrt[1 - c^2]] s + 63  x/256

ReplaceAll[%, {c -> Cos[x], s -> Sin[x]}]

Series[Sin[Sinh[x]]/ArcSin[ArcSinh[x]], {x, 0, 20}]

Series[Sinh[Sin[x]]/ArcSinh[ArcSin[x]], {x, 0, 20}]

h[x_] := f[g[x]];

sh[x_] := h'''[x]/h'[x] - 
        3/2 (h''[x]/h'[x])^2

sh[x]

sf[x_] := f'''[x]/f'[x] - 
          3/2 (f''[x]/f'[x])^2;
sg[x_] := g'''[x]/g'[x] - 
          3/2 (g''[x]/g'[x])^2;

sh[x] - sf[g[x]] g'[x]^2 - sg[x]

Simplify[%]

f[x_] := x/(1 - x)^2

sf[x]

Simplify[%]