Common problems involving derivatives are most often
expressed as
$f\text{'}\left(x\right)=\frac{dy}{dx}$, or the derivative of 'y' WITH REPSECT TO 'x', (the rate
at which 'y' changes AS 'x' CHANGES).

These derivatives are frequently applied to functions where the variable identified in 'RESPECT TO' clause is the only variable in the function (e.g. f(x) = 3x + 2).

But there are instances where a function may be expressed using variables that ARE NOT the same variable stated in the 'WITH RESPECT TO' clause. For example, it may be needed to find the derivative WITH RESPECT TO 'x' of the function $4{y}^{3}$.

Here the function's variable ('y') is not the same variable as in the WITH RESPECT TO 'x' clause. In these cases we apply the chain rule, considering the 'y' as the innermost function. This example further describes how this implicit differentiation works by finding the slope of the tangent line at point (1,-1) for the function:

${x}^{2}+3xy-3{y}^{2}=-4$

These derivatives are frequently applied to functions where the variable identified in 'RESPECT TO' clause is the only variable in the function (e.g. f(x) = 3x + 2).

But there are instances where a function may be expressed using variables that ARE NOT the same variable stated in the 'WITH RESPECT TO' clause. For example, it may be needed to find the derivative WITH RESPECT TO 'x' of the function $4{y}^{3}$.

Here the function's variable ('y') is not the same variable as in the WITH RESPECT TO 'x' clause. In these cases we apply the chain rule, considering the 'y' as the innermost function. This example further describes how this implicit differentiation works by finding the slope of the tangent line at point (1,-1) for the function:

${x}^{2}+3xy-3{y}^{2}=-4$

Implicit differentiation

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