If y = f(x)g(x), then dy/dx = f‘(x)g‘(x). If it is true, explain your answer. If false, provide a counterexample. True or False?

2 Answers
Jun 18, 2018

Answer:

False

Explanation:

Take #f(x)=x^2+1,g(x)=x#
then

#f(x)g(x)=x^3+x#
then
#f'(x)*g'(x)=2x#
But

#(f(x)g(x))'=3x^2+1#
It must be
#(f(x)g(x))'=f'(x)*g(x)+f(x)*g'(x)#

Jun 18, 2018

Answer:

The statement is false .

The product rule provides the correct formulation:

#y = f(x)g(x) => y = f(x)g'(x) + f'(x)g(x) #

Explanation:

We can readily disprove the given statement:

Consider:

#f(x)=x# and #g(x)=x#

Then differentiating wrt #x# we have:

#f'(x)=1# and #g'(x)=1 => f'(x)g'(x)=1#

And #y=f(x)g(x) =x^2 => dy/dx = 2x != f'(x)g'(x)#

And so By counterexample, the statement is false .

In fact the product rule provides the correct formulation:

#y = f(x)g(x) => y = f(x)g'(x) + f'(x)g(x) #