2.4 The Chain Rule Remember the composition of two functions? The chain rule is used when you have the composition of two functions.

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Presentation transcript:

2.4 The Chain Rule Remember the composition of two functions? The chain rule is used when you have the composition of two functions.

f(x) = x3 Find y’ for y = (x2 + 1)3 g(x) = x2 + 1 y’ = 3(x2 + 1)2 (2x) = 6x(x2 + 1)2 Find y’ for y = (3x – 2x2)3 y’ = 3(3x – 2x2)2(3 – 4x) Find f’(x) for

Differentiate rewritten as = -7(2t – 3)-2 g’(t) = 14(2t – 3)-3(2) Differentiate rewritten as Factor

Differentiate rewritten as Bot * Top’ – Top * Bot’ (Bot)2 Quotient Rule

Differentiate

Derivatives of Trigonometric Functions

Applying the Chain Rule to trigonometric functions y = sin 2x y = cos (x – 1) y = tan 3x y = cos (3x2) y = cos2 3x y’ = (cos 2x) (2) = 2 cos 2x y’ = -sin (x – 1) (1) = -sin (x – 1) y’ = sec2 3x (3) = 3 sec2 3x y’ = -sin (3x2) (6x) = -6x sin (3x2) rewritten as y = (cos 3x)2 y’ = 2(cos 3x)1 (-sin 3x) (3) y’ = -6 cos 3x sin 3x

Differentiate rewritten as