<s>[INST] How do I establish the differentiability of this function by definition? Given this function
$f(x) = (x^2 + 1)^{\sin(x)} $
How would I establish its differentiability over the entire function? I understand how to establish differentiability at a point $c$, by assessing the limit of the different quotient at $x = c$. But how would I establish the differentiability of the entire function?
 [/INST] You shouldn't use the definition of derivative, instead you should use a tricky technique:
$\displaystyle f(x)=(1+x^2)^{\sin x} \implies \ln(f(x)) = \ln\left( (1+x^2)^{\sin x} \right) = \sin x \ln(1+x^2)$
$\displaystyle \implies \frac{d}{dx} \ln(f(x)) = \frac{d}{dx} \sin x \ln(1+x^2) \implies \frac{f'(x)}{f(x)} = \cos x \ln(1+x^2) + \sin x \frac{2x}{1+x^2}$
$\displaystyle f'(x)= f(x) \left( \cos x \ln(1+x^2) + \sin x \frac{2x}{1+x^2} \right) = (1+x^2)^{\sin x} \left( \cos x \ln(1+x^2) + \sin x \frac{2x}{1+x^2} \right)$
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