Chain Rule Examples

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Example 1: Find the derivative f '(x), if f is given by
f(x) = 4 cos (5x - 2)
Solution to Example 1
Let u = 5x - 2 and y = 4 cos u, hence du / dx = 5 and dy / du = -4 sin u
We now use the chain rule f '(x) = (dy / du) (du / dx) = - 4 sin (u) (5)
We now substitute u = 5x - 2 in sin (u) above to obtain f '(x) = - 20 sin (5x - 2)

Example 2:
Find the first derivative of f if f is given by
f(x) = sin 2 (2x + 3)
Solution to Example 2
Let u = sin (2x + 3) and y = u 2 , hence du / dx = 2 cos(2x + 3) and dy / du = 2 u
Use the chain rule f '(x) = (dy / du) (du / dx) = 2 u 2 cos(2x + 3)
Substitute u = sin (2x + 3) above to obtain f '(x) = 4 sin (2x + 3) cos (2x + 3)
Use the trigonometric fromula sin (2x) = 2 sinx cos x to simplify f '(x) f '(x) = 2 sin (4x + 6)

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