Functions and are differentiable at with and

MATHEMATICS · Subjective · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Functions

f

and

g

are differentiable at

x=4

with

f(4)=3,\;g(4)=2,\;f'(4)=0,

and

g'(4)=5

. A student claims that the derivative at

x=4

of the product

h(x)=f(x)g(x)

equals

f'(4)g'(4)

. Explain why this claim is incorrect, citing the appropriate relationship between

f,\;g,

and their derivatives.

Related theory

Differentiation Rules
Derivatives of standard functions via chain rule
Derivatives of composite functions
Continuity and Differentiability

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