Prove $\exists\delta>0\ni\forall f\in C([0,1])$ with sup $|f|<\delta\ \exists g\in C^1([0,1])$ such that $g'+g^{30}=f$ (Use Implicit function theorem)

Question

Using Implicit function theorem show that that there exists $\delta>0$ such that
For all $f\in C([0,1];\Bbb{R})$ with $\text{sup }|f|<\delta,\ \exists g\in C^1([0,1];\Bbb{R})$ such that $$g'(t)+g(t)^{30}=f(t)$$

Although I feel like it as a differential equation $\displaystyle{\frac{dy}{dt}+y^{32}=f(t)}$. But I'm not getting any idea how to apply Implicit Function Theorem to get a $\delta>0$. Can anyone give any idea or way out to start with? Thanks for help in advance.

Edit: Later I have thought one way out

Define $T:C([0,1])\times C^1([0,1])\to C^1([0,1])$ by $T(f,g)(s)=g(s)-g(0)+\int\limits_0^s (g^{30}-f)$. Then $T(0,0)=0$, so we have to apply implicit function theorem to this $T$. Here I have taken sup norm for $C([0,1])$ and the $C^1$ norm (defined as $\lVert f\rVert_{C^1}=\text{sup }|f|+\text{sup }|f'|$) with respect to which $C^1$ is banach space (complete). Is it okay?