def _backward_through_llm_chain(variables: List[Variable], 
                                    response: Variable, 
                                    prompt: str, 
                                    system_prompt: str,
                                    backward_engine: EngineLM,
                                    in_context_examples: List[str]=None):
        """
        Backward through the LLM to compute gradients for each variable, in the case where the output has gradients on them.
        i.e. applying the chain rule.
        
        :param variables: The list of variables to compute gradients for.
        :type variables: List[Variable]
        :param response: The response variable.
        :type response: Variable
        :param prompt: The prompt string.
        :type prompt: str
        :param system_prompt: The system prompt string.
        :type system_prompt: str
        :param backward_engine: The backward engine to use for computing gradients.
        :type backward_engine: EngineLM

        :return: None
        """
        for variable in variables:
            if not variable.requires_grad:
                continue

            backward_info = {
                "response_desc": response.get_role_description(),
                "response_value": response.get_value(),
                "response_gradient": response.get_gradient_text(),
                "prompt": prompt,
                "system_prompt": system_prompt,
                "variable_desc": variable.get_role_description(),
                "variable_short": variable.get_short_value(),
                "in_context_examples": "\n".join(in_context_examples) if in_context_examples is not None else [],
            }
            
            backward_prompt = LLMCall_with_in_context_examples._construct_llm_chain_backward_prompt(backward_info)

            logger.info(f"_backward_through_llm prompt", extra={"_backward_through_llm": backward_prompt})
            gradient_value = backward_engine(backward_prompt, system_prompt=BACKWARD_SYSTEM_PROMPT)
            logger.info(f"_backward_through_llm gradient", extra={"_backward_through_llm": gradient_value})
            
            var_gradients = Variable(value=gradient_value, role_description=f"feedback to {variable.get_role_description()}")
            variable.gradients.add(var_gradients)
            conversation = CONVERSATION_TEMPLATE.format(**backward_info)
            variable.gradients_context[var_gradients] = {
                "context": conversation, 
                "response_desc": response.get_role_description(),
                "variable_desc": variable.get_role_description()
            }
            
            if response._reduce_meta:
                var_gradients._reduce_meta.extend(response._reduce_meta)
                variable._reduce_meta.extend(response._reduce_meta)

        Backward through the LLM to compute gradients for each variable, in the case where the output has gradients on them.
        i.e. applying the chain rule.
        
        :param variables: The list of variables to compute gradients for.
        :type variables: List[Variable]
        :param response: The response variable.
        :type response: Variable
        :param prompt: The prompt string.
        :type prompt: str
        :param system_prompt: The system prompt string.
        :type system_prompt: str
        :param backward_engine: The backward engine to use for computing gradients.
        :type backward_engine: EngineLM

        :return: None
        
textgrad/autograd/llm_ops.py