Chain Rule

f(x)

sin(x)

cos(x)

2

x

1

2

x

+1

2

x

+1

2

(1-x)

x



3

x

g(x)

sin(x)

cos(x)

2

x

1

2

x

+1

2

x

+1

2

(1-x)

x



3

x

plot domain

2

plot range

2

x value

1

The derivative of the composition of two functions is given by the chain rule, which states that the derivative of

f(g(x))

is

f'(g(x))g'(x)

. The graphs of

f(g(x))

and its derivative are colored blue and purple. The tangent to

f(g(x))

at the red point is the green line.