Implicit differentiation

In calculus, implicit differentiation is a technique for finding the derivative at a given point on a curve that is not necessarily a true function by taking advantage of implicit functions.[cite:@openstax2020calculus]

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• $\frac{d}{dx}\sin x=\cos x$; $\frac{d}{dx}\sin y=\cos (y\frac{dy}{dx})$.

• The notation is horribly deceptive. Remember that $y$ is a function of $x$. Really, we should not be writing $y$, but $y\left(x\right)$. Realising this, you can apply the chain rule to take derivatives of terms involving $y$.[cite:@lamarimplicit5]

• It is not generally the case that $\frac{dy}{dx}$ can be expressed in terms of $x$ only.[cite:@openstax2020calculus] This does not impede differentiation, as the derivative at a given point is still found by substituting $x$ and $y$.

• Perhaps it should be viewed as a generalised derivative? Differentiating a non-functional relation, yielding a relation?