Implicit differentiation techniques from "summary" of Differential Calculus by S Balachandra Rao
When dealing with functions that are not explicitly expressed in terms of a single variable, we resort to implicit differentiation techniques. This is particularly useful when the given equation involves both dependent and independent variables, making it difficult to isolate the dependent variable explicitly. To differentiate implicitly, we treat the dependent variable as a function of the independent variable, even if it is not explicitly defined as such. This allows us to differentiate both sides of the equation with respect to the independent variable, treating the dependent variable as an unknown function. The chain rule plays a crucial role in implicit differentiation, as it allows us to differentiate composite functions where the dependent variable is nested within another function. This enables us to find the derivative of the dependent variable in terms of the independent variable, even when it is not explicitly given. By differentiating both sides of the equation with respect to the independent variable, we can isolate the derivative of the dependent variable in terms of the independent variable and any other variables present in the equation. This provides us with a way to determine the rate of change of the dependent variable with respect to the independent variable, even in cases where the relationship is not explicitly defined. Implicit differentiation techniques are essential in dealing with complex equations where the dependent variable is not explicitly expressed in terms of the independent variable. By treating the dependent variable as an unknown function and applying the chain rule, we can find the derivative of the dependent variable with respect to the independent variable, allowing us to analyze the relationship between the variables in a more intricate manner.
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