# Derivatives of Implicit Functions

## Derivatives of Implicit functions

If the 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 in the form y = ф(x), then y is said to be an implicit function of x. To find $$\frac{dy}{dx}$$ in such a case, we differentiate both sides of the given relation with respect to x. keeping in mind that the derivative of ф(x), with respect to x is $$\frac{d\phi }{dy}.\frac{dy}{dx}$$.

For example: $$\frac{d}{dx}\left( \operatorname{siny} \right)=\cos y\frac{dy}{dx}$$.

$$\frac{d}{dx}\left( {{y}^{2}} \right)=2y.\frac{dy}{dx}$$.

Differentiation of Parametric Functions: Sometimes x and y are given as functions of a single variable.

Example: x = ф(t), y = Ѱ(t) are two functions and t is a variable. In such a case x and y are called parametric functions or parametric equations and t is called parameter.

To find $$\frac{dy}{dx}$$ in case of parametric functions, we first obtain the relationship between x and y by eliminating the parameter t and then we differentiate it with respect to x. But every time it is not convenient to eliminate the parameter. Therefore, $$\frac{dy}{dx}$$ can also be obtained by the following formula $$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$.

Differentiation of a function with respect to another function: In this section we will discuss derivative of a function with respect to another function. Let u = f(x) and v = g(x) be two functions of x. Then to find the derivative of f(x) with respect g(x) i.e., to find $$\frac{du}{dv}$$ we use the following formula $$\frac{du}{dv}=\frac{\frac{du}{dx}}{\frac{dv}{dx}}$$.

Thus, to find the derivative of f(x) with respect to .g(x), we first differentiate both with respect to x and then divide the derivative of f(x) with respect to x .by the derivative of g(x) with respect to x.

Example: Find the derivative of log sinx

Solution: Let f(x) = log sinx

$$f’\left( x \right)=\frac{1}{\sin x}.\frac{d}{dx}\left( \sin x \right)$$,

$$=\frac{\cos x}{\sin x}$$,

= cotx.