Is the differential of a surjection a surjection?

Question

Let $\phi:M \to N$ be a smooth map. Let $\text d\phi:TM\to TN$ denote its differential. If $\phi$ is surjective then is $\text d\phi$ surjective?

I don't know much about differential geometry. I don't have a counterexample, and it seems like a plausible statement

Answer 1

If you define $\phi\colon\Bbb R\longrightarrow\Bbb R$ by $\phi(x)=x^3$, then $\phi$ is a smooth surjective map. However, $\mathrm d\phi$ is not surjective at $0$, since it is the null function.