Is $f(x)=e^x \cdot \cos(e^x)$ a tempered distribution? Let $f(x)=e^x \cdot \cos(e^x)$. Define $$T_f(\varphi)=\int_{-\infty}^{+\infty} f(x) \cdot \varphi(x) \ .$$ I would like to know if $T_f$ defined with the formula above defines a tempered distribution (in the sense of the definition given here#Tempered_distributions_and_Fourier_transform)).

Hint: The integral of $f(x):=e^x \cdot \cos(e^x)$ is $F(x):=\sin(e^x)$ and defines a function of "slow growth" for $x\in\mathbb{R}$. Deduce that the derivative is a tempered distribution (without being a function of slow growth itself).