Update vercel-ai-sdk.mdx

add instructions for how to use google ai studio w/ vercel ai sdk

Author: kathayl

src/content/docs/ai-gateway/integrations/vercel-ai-sdk.mdx

@@ -35,6 +35,18 @@ const anthropic = createAnthropic({
 });
 ```
 
+### Google AI Studio
+
+If you're using the Google AI Studio provider in AI SDK, you need to append `/v1beta` to your Google AI Studio-compatible AI Gateway URL to avoid errors. The `/v1beta` path is required because Google AI Studio's API includes this in its endpoint structure, and the AI SDK sets the model name separately. This ensures compatibility with Google's API versioning.
+
+```typescript
+import { createGoogleGenerativeAI } from '@ai-sdk/google';
+
+const google = createGoogleGenerativeAI({
+	baseURL: `https://gateway.ai.cloudflare.com/v1/{account_id}/{gateway_id}/google-ai-studio/v1beta`,
+});
+```
+
 ### Other providers
 
 For other providers that are not listed above, you can follow a similar pattern by creating a custom instance for any AI provider, and passing your AI Gateway URL. For help finding your provider-specific AI Gateway URL, refer to the [Supported providers page](/ai-gateway/providers).