W n 4.92 -4.56 Td 7.92 0 Td 2195 3032.67 348 6 re <>stream [(a)-0.588548(\))-3.44553( )-386.843(f)8.57043(\()-3.44548(x)7.34787(\))-3.44548( )-2.33439(=)-0.748061( )250]TJ Ableitung von trigonometrische Funktionen Aufgaben mit Lösungen Author: Sascha Frank Subject: Ableitung Keywords: Ableitung, Aufgaben, Lösungen, trigonometrisch, Funktionen Created Date: 12/26/2016 7:49:35 PM /R15 9.96 Tf S Q 59.16 4.56 Td ( )Tj (x)Tj ()Tj 488.16 0 Td /R7 9.98676 Tf f 0.12 0 0 -0.12 95.76 145.68 Tm /R7 9.975 Tf /R15 9.96 Tf q ( )Tj /R7 9.98676 Tf W n [(b)-5.26316(\))-3.84214( )-327.444(f)8.18771(\()-3.84237(x)6.76692(\))-3.84237( )-2.63158(=)-1.41345( )250]TJ 1 0 0 1 50.4 58.2802 Tm h f /R7 9.98676 Tf [( )-2.33512(0)-4.67024( )]TJ /R7 9.975 Tf (l)Tj /H 30 ( )Tj T*[(A)-0.891628(u)1.78326(f)-4.34967(g)-6.0241(a)-6.0241(b)1.78326(e)-1.78326( )-3.01205(2)-6.02416(:)-4.34967( )-3.01205(P)-3.45768(r)-1.78326(o)-6.0241(d)1.78326(u)1.78252(k)13.8307(t)-4.34894(f)-4.34894(o)-6.0241(r)-1.78252(m)25.771( )]TJ 2898 3569.67 6 6 re /R7 9.98532 Tf /R8 9.96 Tf 3 0 obj W n /R7 9.98676 Tf Q /R15 9.96 Tf ()Tj T*[(A)-0.891628(u)1.78326(f)-4.34967(g)-6.0241(a)-6.0241(b)1.78326(e)-1.78326( )-3.01205(9)-6.02416(:)-4.34967( )-3.01205(S)1.78326(y)-6.0241(m)25.7708(m)25.7715(e)-1.78252(t)-4.34894(r)-1.78252(i)0.89126(e)-1.78319( )]TJ q 0.952057 0 0 -1.04794 0 0 cm ET /R7 9.98676 Tf /R7 9.975 Tf /R36 9.99516 Tf f ( )Tj 1 0 0 1 383.28 390.32 Tm 147.72 -1.56 Td /R7 9.98676 Tf 51.6 0 Td 883 5155.67 l 1 0 0 1 155.52 385.28 Tm 2.52 0 Td 2.52 0 Td /R7 10.0134 Tf 7.92 0 Td 2898 3957.67 6 6 re /R7 9.98676 Tf ()Tj f ()Tj (\()Tj 3.24 -4.56 Td ET Lösungskarte 4.92 -4.56 Td 1 0 0 1 320.64 320.48 Tm 37.68 -1.2 Td 2.52 0 Td 7.44 4.56 Td 3270 3326.67 l ()Tj 1 0 0 1 417.6 468.32 Tm 20.64 4.56 Td 5 0 obj /W 48 20 0 obj 8.33333 0 0 8.33333 0 0 cm BT f endobj 2549 4257.67 349 6 re [(1)-4.00788( )-2.00394(u)7.99232(n)7.99232(d)-4.00788( )-2.00394(P)-8.00989(\()-3.00591(0)500.008]TJ 7.92 0 Td Q ( )Tj (6)Tj 11.64 0 Td /R7 10.0134 Tf 4.128 w 1 0 0 1 118.08 397.64 Tm /R7 9.96 Tf [( )-2.33512(2)-4.66731(\))-3.44695(\()-3.44695(x)7.3486( )250]TJ /R7 9.98676 Tf Q f 2549 4157.67 349 6 re 4.92 -4.56 Td (2)Tj [(\()-2.55127(x)500]TJ (2)Tj 87.36 0 Td /R7 9.98676 Tf [(l)1.59609(d)-4.74134(e)-0.651347(r)-3.49145( )-134.565(i)1.59609(n)7.27631( )-134.565(e)-0.651347(i)1.59609(n)7.27631( )250]TJ 13.44 -6.36 Td /R7 9.98676 Tf 3856 1069.67 l 43 0 obj [(i)0.89126(\))-4.34944( )-557.229(f)]TJ Q 2975 4608.67 l 390 2632.67 618 6 re 3613 5421.67 l [( )-3.01205(1)-6.0241(\))-4.34944( )-12231.9(k)6.0241(\))-4.34944( )-340.361(f)]TJ h /R7 9.975 Tf Als letzten Schritt betrachtet man die Schnittpunkte mit der -Achse. /R7 6.48 Tf (1)Tj ()Tj /R7 6.48 Tf ()Tj (4)Tj 1598 3857.67 6 6 re /R7 9.98532 Tf [( )-2.00394(+ )-2.00394(7)-4.00788(x)7.99232( )250]TJ 7.56 -7.32 Td [(0)-4.66731(\))-3.44695( )-2.33512(u)7.3486(n)7.3486(d)-4.66731( )-2.33512(P)556]TJ 8.4 0 Td /R15 9.96 Tf /R7 9.98676 Tf Q /R7 9.98676 Tf 7.92 0 Td 7.44 0 Td f /R15 9.96 Tf ()Tj ()Tj 8.04 -7.32 Td /R7 9.99984 Tf 5.28 4.56 Td q q (2)Tj ()Tj /R7 6.48 Tf /R7 9.98676 Tf 7.8 0 Td /R7 6.48 Tf 3.24 -4.56 Td (x)Tj q 3.24 -4.56 Td (,)Tj /R15 9.96 Tf 2997 4045.67 l -60.84 -12.36 Td 3875 1160.67 l q [(\()-3.44695(1)500]TJ ( )Tj 36.36 0 Td 1598 3475.67 6 94 re /R7 9.99984 Tf [( )-1.66184(t)2.36987( )-1.66184(=)0.754848( )-1.66184(x)500]TJ 2898 4894.67 6 6 re /R7 6.48 Tf S (1)Tj /R7 9.98676 Tf ET 12.48 4.56 Td 8.04 -1.56 Td q Q [(+)0.754848( )-1.6633(1)-3.32561( )]TJ 2731 2531.67 l (3)Tj 2898 4057.67 6 6 re [( )-2.00394(6)-4.00788(\))-3.00591( )-2.00394(:)1.99515( )-2.00394(\()-3.00591(x)7.99232( )-2.00394(+ )-2.00394(2)-4.00788(\))-3.00539( )]TJ 4.92 4.56 Td 2917 1934.67 m ()Tj 1692 5946.67 l ET 384 2532.67 6 6 re [(0)-4.66731(\))-3.44695( )-2.33512(u)7.3486(n)7.3486(d)-4.66731( )-2.33512(P)556]TJ 11.04 TL [( )-2.00394(+ )-2.00394(7)-4.00788(x)500.008]TJ (t)Tj ID &��f� `���v�('���@��(7؂���q��[���a��� �`�olO��]�x`� /R7 9.98676 Tf [()-2106.24()-2526.85()-1685.62()777.999]TJ Q 3.24 1.56 Td 154.08 -1.56 Td q [(\()-3.44695(x)7.3486(\))-3.44695( )-2.33512(=)-0.748061( )-2.33512(x)500]TJ ��5�Sљ�|����26ӵz��D�U�U2�߂X)g�!��� �%���{ 4�h y�'���߇U�2�Œ�K�[ W n [(a)-1.11277(\))-3.84214( )-387.594(f)8.18771(\()-3.84237(x)6.76692( )-2.63158(+)-1.41345( )-2.63158(1)-5.26316(\))-3.84237( )-2.63158(=)-1.41345( )-2.63158(x)500]TJ 390 3569.67 618 6 re 2189 4788.67 6 6 re 1 0 0 1 322.8 538.64 Tm [( )-2.37067(+)-0.830321( )-2.37067(\()-3.49439(t)277.999]TJ 1008 2738.67 6 94 re 1 0 0 1 153 442.04 Tm – Ableitungen ganzrationaler Funktionen berechnen – Tangentensteigungen berechnen – Nullstellen berechnen – Extremstellen mit dem Vorzeichenwechselkriterium bestimmen – Gleichungssysteme lösen k Check-in: ob Sie die oraussetzungen 6. q ET 1 0 0 1 299.64 377.48 Tm <>stream 7.92 0 Td 33.96 -1.56 Td 5 0 obj 7.92 0 Td 8.04 -1.56 Td 38.16 0 Td 2898 3963.67 6 94 re Q 1008 3038.67 6 94 re 1 0 0 1 335.76 314.36 Tm 11.4 TL -288.6 -19.44 Td /R7 9.98676 Tf 59.2 0 0 0 60 34 d1 /R7 9.96 Tf 4.92 -4.56 Td /R7 6.48 Tf /R7 9.99984 Tf 9.96 4.56 Td /R7 6.48 Tf /R7 9.975 Tf ( )Tj 3247 4651.67 m /R7 9.96 Tf (1)Tj q <> 2189 4257.67 6 6 re /R15 9.96 Tf /R15 9.96 Tf 3.24 -4.56 Td S ( )Tj /R7 9.975 Tf /R7 9.98532 Tf <>stream /R36 9.99516 Tf 0.12 0 0 -0.12 118.08 589.8 Tm [(1/)0.221535(2)500]TJ q [(W)-5.25696(e)-0.588731(l)1.634(c)-0.588915(h)7.34787(e)-0.588915( )-2.33439(W)-5.25696(e)-0.588915(r)-3.44548(t)1.634(e)-0.588915( )-2.33439(d)-4.66805(�)7.34787(r)-3.44548(f)8.57043(e)-0.588181(n)7.34713( )-2.33366(d)-4.66878(i)1.634(e)-0.588181( )-2.33366(K)1.04582(o)-4.66878(e)-0.588181(f)8.57043(f)8.57043(i)1.634(z)-0.588181(i)1.634(e)-0.586714(n)7.34713(t)1.634(e)-0.588181(n)7.34713( )-2.33366(a)-0.588181(,)-2.33366( )-2.33366(b)-4.66878( )-2.33366(u)7.34713(n)7.34713(d)-4.66878( )-2.33366(c)-0.589648( )-2.33512(a)-0.589648(n)]TJ [()-3320.02()-3620.46()-3572.39()777.999]TJ 8.33333 0 0 8.33333 0 0 cm BT ET [( )-2.00394(8)-4.00788(x)500.008]TJ ()Tj ET /R7 6.48 Tf /R15 9.96 Tf /R7 6.48 Tf /R7 10.0134 Tf [( )-2.37067( )]TJ 0] 23.28 0 Td 1289 4735.67 m -0.6 15.6 Td 145.08 -1.56 Td q 33.12 0 Td 837 4258.67 m /R7 6.48 Tf 1 0 0 1 123.24 565.04 Tm (3)Tj 4602.66 -3087.64 m ()Tj 8.256 w /R7 9.98676 Tf EI 1 0 0 1 380.76 468.32 Tm /R15 9.96 Tf /D[1 f 3.24 1.56 Td [(4 )250]TJ (2)Tj 1228 5721.67 l q (1)Tj /R7 6.48 Tf /R7 9.98532 Tf ( )Tj [(4)-1074.08(t)278]TJ [( )-2.63158(t)1.30698( )-2.63158(+)-1.41271( )-2.63158(1)-5.26316(\))-3.84164(\()-3.84164(x)6.76692( )250]TJ /R7 6.48 Tf ()Tj z2 < … q 1398 4139.67 m ()Tj 8.33333 0 0 8.33333 0 0 cm BT [( )-2.33512(2)-4.66731(t)1.634(\))-3.44695(x)500]TJ 8.33333 0 0 8.33333 0 0 cm BT [(\()-3.84164(x)500]TJ Q (a)Tj (2)Tj 117.96 -1.56 Td 1681 4045.67 l /R15 9.96 Tf 3517 4189.67 m 30 0 obj /R15 9.96 Tf (2)Tj /R7 6.48 Tf 7.92 0 Td 1 0 0 1 95.16 466.76 Tm 8.33333 0 0 8.33333 0 0 cm BT 237 0 Td 3785 3326.67 l [(\()-3.49365(x)7.27631(\))-3.49365( )-2.37067(=)-0.829587( )-2.37067(4)-4.74134(t)1.59389(x)500]TJ 2543 3038.67 6 94 re W n 83 0 0 0 83 30 d1 [( )-2.33366(2)-4.66829( )-7308.01(f)8.56896(\))-3.44696( )-507.003(f)8.56896(\()-3.44695(x)7.3486(\))-3.44695( )-2.33512(=)-0.748061( )-2.33512(t)1.634(x)500]TJ 2890 2026.67 l 10.2 4.56 Td ET (2)Tj 19.44 TL /R7 9.96 Tf /R15 9.96 Tf f W n 1598 2732.67 6 6 re (\()Tj [(e)-0.589648(\))-3.44696( )-386.844(x)]TJ ()Tj ()Tj /R7 6.48 Tf 8.33333 0 0 8.33333 0 0 cm BT f (4)Tj 33 3 Td [(4)-5439.91(3)-5697.43(2)500]TJ S 2777 4842.67 l Aufgaben Aufgaben rechnen; Stoff Stoff ansehen (+Video) 3868 4651.67 l -417.48 -13.92 Td ()Tj [(\()-3.44695(0)500]TJ [(+)-2.26492( )-3.01205(0)-6.0241(,)-3.01205(5)-6.0241(x)500]TJ 3.72 0 Td 817 2308.67 l ( )Tj 8.33333 0 0 8.33333 0 0 cm BT -355.2 -11.28 Td 3323 3235.67 l [(c)-1.11277(\))-3.84214( )-387.594(f)]TJ 18.96 -1.56 Td ()Tj ()Tj /R7 6.48 Tf /R7 6.48 Tf 1.8 1.56 Td 7.92 0 Td /R7 6.48 Tf 23.16 4.56 Td /R15 9.96 Tf 7.92 0 Td 0.12 0 0 -0.12 95.16 133.32 Tm 8.33333 0 0 8.33333 0 0 cm BT 12.36 TL 3.24 -4.56 Td f /R15 9.96 Tf [(\()-3.44695(x)7.3486(\))-3.44695( )-2.33512(=)-0.748061( )-2.33512(\()-3.44695(x)7.3486( )250]TJ 16 0 obj 8.33333 0 0 8.33333 0 0 cm BT 1 0 0 1 319.8 275.84 Tm ()Tj (2)Tj /R7 9.98676 Tf 20.64 4.56 Td Q 0 14.28 Td /R15 9.96 Tf 7.92 0 Td [(A)-0.891628(u)1.78326(f)-4.34967(g)-6.0241(a)-6.0241(b)1.78326(e)-1.78326( )-3.01205(6)-6.02416(:)-4.34967( )-3.01205(N)-0.891996(u)1.78326(l)0.89126(l)0.89126(s)3.45768(t)-4.34894(e)-1.78252(l)0.89126(l)0.89126(e)-1.78252(n)1.78252(b)1.78252(e)-1.78252(s)3.45768(t)-4.34894(i)0.89126(m)25.771(m)25.7715(u)1.78252(n)1.78252(g)-6.0241( )-3.01205(d)1.78252(u)1.78252(r)-1.78252(c)-1.78252(h)1.78252( )-3.01205(S)1.78252(u)1.78252(b)1.78252(s)3.45915(t)-4.35041(i)0.89126(t)-4.35041(u)1.78252(t)-4.35041(i)0.89126(o)-6.0241(n)]TJ /R7 6.48 Tf f /R7 9.975 Tf 3.24 -4.56 Td /R7 9.99984 Tf (t)Tj W n 1.8 1.56 Td ET Q 7.92 0 Td S Bestimmen Sie (1) jeweils eine sinnvolle, zusammenhängende Definitionsmenge (2) Nullstellen (3) Symmetrie (4) Verhalten der Steigung an den Grenzen des jeweiligen Definitionsbereiches. Q 1 0 0 1 92.64 651.68 Tm 29.4 0 Td 3.72 0 Td 2658 3095.67 m /R7 9.975 Tf 2.52 0 Td �F))���� l,��v��OV���2z�ExPRvj(���^+�p�֠@o(�?�p.�\�H��,�ǀo:9��9@�',��]�F $�N�����y����ɏo�@�����=('���Qt�he�`�"UY���O�V��R-��Pu�����@� 4�&�!�A��Һ��!=�ʯ�NM7���Kߢ� O�dum^�� ń��v��У�H��#5}�t�99)�$�v����w�0E���{���6�d'�"�a�϶t8K �}�%��a�a%ZN�{x�}UF��1�T�ƴ.ت�e/J�#�)�� 2898 4263.67 6 94 re /R7 9.96 Tf [( )-2.33366(1)-4.66829( )-4904.83(l)1.634(\))-3.44696( )-555.067(f)]TJ f /R15 9.96 Tf f /R15 9.96 Tf /R15 9.96 Tf ()Tj /R7 9.98532 Tf /R7 9.975 Tf (t)Tj [(2 )250]TJ (5)Tj ()Tj (2)Tj -379.68 -12.36 Td [(f)8.57043(\))-3.44553( )-507.002(f)8.57043(\()-3.44548(x)7.34787(\))-3.44548( )-2.33439(=)-0.748061( )250]TJ (t)Tj /R15 9.96 Tf 4496 2619.67 l Q ( )Tj q 1 0 0 1 115.44 372.92 Tm [(a)-0.588548(\))-3.44553( )-386.843(f)8.57043(\()-3.44548(x)7.34787(\))-3.44548( )-2.33439(=)-0.748061( )-2.33439(\()-3.44548(x)7.34787( )-2.33439(+)-0.748061( )-2.33439(3)-4.66805(\))-3.44548(\()-3.44548(x)7.34713( )-2.33366(+)-0.748061( )-2.33366(2)-4.66878(\))-3.44548(\()-3.44548(x)7.34713( )-2.33366(+)-0.748061( )-2.33366(1)-4.66878(\))-3.44553( )-10263.9(h)7.3486(\))-3.44696( )-338.781(f)8.56896(\()-3.44695(x)7.3486(\))-3.44695( )-2.33512(=)-0.748061( )250]TJ 13.2 4.56 Td /R7 9.975 Tf 8.33333 0 0 8.33333 0 0 cm BT [( )-1.66184(+)0.754848( )-1.66184(5)-3.3266(t)278]TJ /R15 9.96 Tf 58.8 0 Td [( )-6859.77(1)-5.26316(,)-2.63158(8)-5.26316(6)-5.26316(0)-5.26316(9)-5.26316(3)-5.26316(7)-5.26316(5)-5.26321( )-2553.01( )-4020.68( )]TJ q 0.952057 0 0 -1.04794 0 0 cm h 384 4057.67 6 6 re /R7 9.975 Tf 1 0 0 1 243.24 175.16 Tm 4.92 4.56 Td f 8.33333 0 0 8.33333 0 0 cm BT 1275 920.668 l Q endobj Q 2543 4463.67 6 94 re /R7 9.99984 Tf [(4)-4.00788(\))-3.00539( )]TJ f 1604 3132.67 585 6 re [( )-2.00394(= )-2.00394(2)-4.00788(,)-2.00394( )-2.00404(x)]TJ /R7 9.98676 Tf 1 0 0 1 93 253.04 Tm Q 1008 2832.67 6 6 re 8.33333 0 0 8.33333 0 0 cm BT (x)Tj 7.92 0 Td 7.92 0 Td 1344 1014.67 l 5.04 0 Td [( )-2.37067(1)-4.74134(2)-4.74134(t)1.59463( )-2.37067(f)8.52326(�)7.27631(r)-3.49439( )-2.37067(t)1.59463( )250]TJ ()Tj 1.8 1.56 Td /R7 6.48 Tf ()Tj [(i)0.89126(n)1.78252( )-231.928(Wo)-6.0241(r)-1.78252(t)-4.35041(e)-1.78252(n)556]TJ /R7 6.48 Tf /R7 6.48 Tf 1 0 0 1 95.04 234.32 Tm h 1260 1357.67 m >> 1 0 0 1 118.92 288.2 Tm /R7 9.98676 Tf /R7 6.48 Tf 11.4 TL 1 0 0 1 98.4 480.68 Tm 8.33333 0 0 8.33333 0 0 cm BT 2189 4057.67 6 6 re [( )-280.121(u)6.0241(n)6.0241(d)-6.0241( )-280.121(s)3.45915(k)6.0241(i)0.894202(z)-1.78252(z)-1.78252(i)0.894202(e)-1.78252(r)-4.34747(e)-1.78319( )-280.121(i)0.894202(h)6.0241(r)-4.34747(e)-1.78252( )250]TJ q 12.12 TL (1)Tj ()Tj q 0.952057 0 0 -1.04794 0 0 cm q /R7 9.98676 Tf /R7 6.48 Tf 18.96 -1.56 Td 3.24 1.56 Td [(\()-3.84164(1)500]TJ 18.48 4.56 Td W n 0.12 0 0 -0.12 104.16 300 Tm /R7 6.48 Tf /R7 9.98676 Tf ( )Tj (4)Tj ()Tj /R7 6.48 Tf 20.64 4.56 Td 13.44 4.56 Td (2)Tj 4.92 -4.56 Td [(\()-4.35041(x)6.0241(\))-4.35041( )-3.01205(=)-2.26492( )-3.01205(\()-4.35041(x)6.0241( )250]TJ 1 0 0 1 371.4 385.28 Tm 3.72 0 Td h ()Tj h /R7 9.98676 Tf q 0.12 0 0 0.12 0 0 cm ( )Tj /R7 9.99984 Tf Q /R15 9.96 Tf f f [( )-2.33512(4)-4.66731(\))-3.44696( )]TJ 7.92 0 Td ()Tj f Q /R7 9.98676 Tf /R7 9.98532 Tf q /R7 9.975 Tf /R22 9.98532 Tf 2760 1069.67 l q f 28.68 4.56 Td /R7 9.975 Tf 384 2538.67 6 94 re /R7 9.96 Tf /R15 9.96 Tf q 0.952057 0 0 -1.04794 0 0 cm f ()Tj f /R7 10.0134 Tf /R7 9.98532 Tf 4.128 w /R7 9.975 Tf [(a)-6.0241(l)0.89126(b)1.78252(i)0.89126(e)-1.77958(r)-1.78252(u)1.78252(n)1.78252(g)-6.0241(s)3.45915(v)-6.0241(e)-1.78252(r)-1.78252(f)-4.35041(a)-6.0241(h)1.78252(r)-1.78252(e)-1.78252(n)1.78012( )]TJ /R7 9.99984 Tf 2898 3463.67 6 6 re W n Q 1008 3032.67 6 6 re 3.24 -4.56 Td 3.24 1.56 Td [(])-3.84164( )]TJ 1 0 0 1 498.48 455.96 Tm 7.8 0 Td 2189 3238.67 6 225 re ET 2733 2490.67 l [(0)-5.26316(,)-2.63158(2)-5.26316(5)-5.26316(6)-5.26316(2)-5.26316(5)500]TJ /R15 9.96 Tf /R7 9.98676 Tf [( )-2.33366(+)-0.748061( )-2.33366(4)-4.66878(x)7.34713( )-2.33366(+)-0.748061( )-2.33366(4)-4.66829( )-10864.7(j)-10.3819(\))-3.44696( )-543.051(f)]TJ 1 0 0 1 378.36 545.6 Tm /R15 9.96 Tf Q /R7 6.48 Tf 10.8 0 Td 8.33333 0 0 8.33333 0 0 cm BT (5)Tj 4337 4560.67 l [(\()-3.44695(0)500]TJ ; Funktionen zeichnen; Schnittpunkte; Funktionsgleichungen; Steigung. 4530 559.668 m Q (t)Tj 4531 465.668 l [(\()-4.35041(x)6.0241(\))-4.35041( )-3.01205(=)-2.26492( )-3.01205(x)500]TJ 1 0 0 1 321.96 466.76 Tm [(0)-6.0241(,)-3.01205(2)-6.0241(5)-6.02416( )]TJ 1.8 1.56 Td Q 2980 4830.67 m ET h 1 0 0 1 464.52 478.04 Tm /R7 9.98676 Tf /R7 9.98676 Tf 1604 3957.67 585 6 re 7.947 w /R15 9.96 Tf Q [( )-2.37067(4)-4.74102( )-10073.2(l)1.59316(\))-3.49439( )-555.182(f)]TJ /R15 9.96 Tf 4 0 obj ()Tj q 8.33333 0 0 8.33333 0 0 cm BT 3.24 1.56 Td 7.92 0 Td 8.33333 0 0 8.33333 0 0 cm BT 8.33333 0 0 8.33333 0 0 cm BT /R7 9.975 Tf
2020 ganzrationale funktionen nullstellen aufgaben mit lösungen pdf