Consider two differentiable functions and with and

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Consider two differentiable functions

g

and

f

with

u=g(x)

and

y=f(u)

. The composite function is

y=f(g(x))

. (a) Using the idea of local linearization, describe qualitatively how a small increment

\Delta x

x

propagates through

u

and then through

y

. In your explanation, refer to the roles of

g'(x)

and

f'(u)

. (b) Explain why the instantaneous rate of change of

y

with respect to

x

at a point

x=a

can be viewed as the product of the instantaneous rate of change of

y

with respect to

u

evaluated at

u=g(a)

and the instantaneous rate of change of

u

with respect to

x

evaluated at

x=a

. Your answer should reference the limit definition of derivative, but a full algebraic proof is not required.