How to find the Gateaux derivative when F(a)=b

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How can i approach finding the Gateaux-derivative $F'(a)(r)$ where $F(a)=b$? I tried $F'(a)(r)=\frac{F(a+rt)-F(a)}{t}=\frac{F(a+rt)-b}{t}$ but I'm not sure how to continue or how to simplify this.