Here are the essential concepts you must grasp in order to answer the question correctly.
Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for equations involving products or compositions of functions.
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Chain Rule
The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This is essential in implicit differentiation when dealing with products of variables.
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Trigonometric Functions
Trigonometric functions, such as sine and cosine, are fundamental functions in calculus that relate angles to ratios of sides in right triangles. In the context of implicit differentiation, understanding how to differentiate these functions is crucial, as they often appear in equations involving angles and products of variables. For example, the derivative of sin(u) is cos(u) multiplied by the derivative of u, which is applied when differentiating equations like sin(xy).
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