Question

give an example of a function that has the given properties in each case.

1) function involves sin(x), and the derivative is gotten using product rule.
2) function involves sin(x), derivative is gotten using chain rule.
3) function involves e^x, and its derivative requires two applications of the chain rule.

Answer 1

x sin(x)

sin(x^2)

sin(5e^(6x))

Answer 2

could you please explain the last one?

Answer 3

there are of course infinitely many answers

Answer 4

but if you want an example that uses the chain rule twice you should come up with a function that involves three composition.

Answer 5

so shouldn't it be sin(5e^6)^x then?

Answer 6

for example
\[f(x)=\sqrt{x}\]
\[g(x)=\cos(x)\]
\[h(x)=e^x\] then
\[h(g(f(x)))=h(g(\sqrt{x}))=h(\cos(\sqrt{x})=e^{\cos(\sqrt{x}}))\]

Answer 7

or
\[f(x)=6x\]
\[g(x)=5e^x\]
\[h(x)=\sin(x)\] so
\[h(g(f(x)))=\sin(5e^{6x})\]

Answer 8

but does that require two applications of the chain rule?

Answer 9

nvm, I get it.