Diagram 1, Black invades move-1 is a very common attack against the joseki when White tenuki. Δ marked stone has alternative at A as well.
Diagram 2, White move-2 kosumi is not the best answer here. Black move-3 & move-5 can connects the group to move-7. White cannot counter-attack with move-8 because Black can atari at A, White kills at B and Black double atari at C.
Diagram 3, White move-1 is best answer here. If Black simply connects back to with move-6. White is thick.
Diagram 4, Black extends down move-1 is great move being calm and simply connect easily with move-3 or the other side.
Diagram 5, If White tries to prevent connection with move-2, it wouldn’t work as Black can easily fight with move-3.
Diagram 6, White move-4 may seems like a strong move. Black can move-7 atari from outside and forces White’s group to lack of efficiency.
Diagram 7, Black move-3 wedges seem great. if White pulled back with move-4, Black easily gains too much at move-7 and White leaves a cutting at A.
Diagram 8, If White atari move-4, Black can easily take White’s one stone down to the right.
Diagram 9, If White atari move-6, Black gets a shape outside but White happily settled down.
Diagram 10, If White’s extension’s move is at 4th-line and when Black invades move-1, White easily acquire wall with move-2 to move-6.
Diagram 11, Black invades at move-1 is the strongest. If White move-2 Black can easily split the group with move-3.
Diagram 12, White connects with move-2 is the only choice. Black move-5 to move-9 does not work because White still get connected and Black attack is a failure.
Diagram 13, If Black move-5 to connect back, White capturing move-6 is too big.
Diagram 14, Thus, Black move-5 atari is the correct sequence then only connecting back with move-7 and ended with nice shape at move-9.
Diagram 15, Hence, White’s move-8 must be played to trade with Black move-9 to escape. White can easily settle the shape with move-12 & move-14.