Differentiation of Implicit Function - Practice Questions & MCQ

Differentiation of Implicit Function

If variables $x$ and $y$ are connected by a relation of the form $f(x, y)=0$ and it is not possible or convenient to express y as a function of x i.e., in the form $\mathrm{y}=\Phi(\mathrm{x})$, then y is said to be an implicit function.

To find $\frac{d y}{d x}$ in such a case, we differentiate both sides of the given relation concerning x keeping in mind that the derivative of $\Phi(\mathrm{y})$ concerning x is $\frac{d \phi}{d y} \times \frac{d y}{d x}$.

For example

$
\frac{d}{d x}(\sin y)=\cos y \frac{d y}{d x}, \frac{d}{d x}\left(y^2\right)=2 y \frac{d y}{d x}
$
It should be noted that $\frac{d}{d y}(\sin y)=\cos y$ but $\frac{d}{d x}(\sin y)=\cos y \frac{d y}{d x}$