Calculus 3-7 CHAIN RULE

Chain Rule If 𝑓 and 𝑔 are differentiable, then the composite function 𝑓∘𝑔 𝑥 =𝑓 𝑔 𝑥 is differentiable and 𝑓 𝑔 𝑥 ′ = 𝑓 ′ 𝑔 𝑥 𝑔 ′ 𝑥

𝑓 𝑥 = 𝑥 4 +1 𝑓 𝑥 = 𝑔(𝑥) 𝑔 𝑥 = 𝑥 4 +1 𝑓 ′ 𝑥 = 1 2 𝑔 𝑥 − 1 2 𝑔 ′ 𝑥 𝑓 ′ 𝑥 = 4 𝑥 3 2 𝑥 4 +1

Leibniz Notation 𝑑𝑦 𝑑𝑥 = 𝑓 ′ 𝑢 𝑔 ′ 𝑥 = 𝑑𝑓 𝑑𝑢 ∗ 𝑑𝑢 𝑑𝑥

𝑓 𝑥 = 8 𝑥 4 +5 3 𝑓 𝑢 = 𝑢 3 𝑓 ′ 𝑢 =3 𝑢 2 𝑢=𝑔 𝑥 =8 𝑥 4 +5 𝑔 ′ 𝑥 =32 𝑥 3 𝑓 ′ 𝑥 = 𝑓 ′ 𝑢 𝑔 ′ 𝑥 𝑓 ′ 𝑥 = 8 𝑥 4 +5 3 (32 𝑥 3 )

𝑓 𝑥 = tan (4−3𝑥) sec (3−4𝑥) 𝑓 ′ 𝑥 =𝑔 𝑥 ℎ ′ 𝑥 +ℎ 𝑥 𝑔 ′ 𝑥 Product Rule 𝑔 𝑥 = tan (4−3𝑥) ℎ 𝑥 = sec (3−4𝑥) 𝑔 𝑢 = tan (𝑢) 𝑢=4−3𝑥 𝑔 ′ 𝑢 =𝑢′ sec 2 𝑢 𝑢 ′ =−3 𝑔 ′ 𝑥 =−3 sec 2 (4−3𝑥)

𝑓 𝑥 = tan (4−3𝑥) sec (3−4𝑥) ℎ 𝑣 = sec (𝑣) 𝑣=3−4𝑥 ℎ ′ 𝑣 =𝑣′ sec (𝑣) tan (𝑣) 𝑣 ′ =−4 ℎ ′ 𝑥 =−4 sec 3−4𝑥 tan 3−4𝑥 𝑓 ′ 𝑥 = tan 4−3𝑥 −4 sec 3−4𝑥 tan 3−4𝑥 + sec (3−4𝑥) −3 sec 2 4−3𝑥

𝑓 𝑥 = sin ( 𝑥 2 +4𝑥) 𝑓 𝑢 = sin 𝑢 𝑓 ′ 𝑢 =𝑢′ cos 𝑢 𝑢= 𝑥 2 +4𝑥 𝑢 ′ =2𝑥+4 𝑓 ′ 𝑥 = (2x+4)cos ( 𝑥 2 +4𝑥)

Problems 3.7 #29-61 odd, 73, 75