Question

What is $2xy$ differentiated implicitly?

Answer 1

Hint: The method of differentiating an implicit equation with respect to the desired variable $x$ thus treating the other variables as unspecified functions of $x$ is known as implicit differentiation.

Complete step-by-step solution:
A partial derivative of a multivariable function is its derivative with respect to one of the variables while the others are kept constant (as opposed to the total derivative, in which all variables are allowed to vary).
The partial derivative of with respect to $x$ is denoted by either $\dfrac{{\partial f}}{{\partial x}}$ or ${f_x}$. Given a function of two variables, $f\left( {x,y} \right)$, the derivative with respect to $x$ only (treating $y$ as a constant) is called the partial derivative of with respect to $x$.
The given function is $f\left( {x,y} \right)\, = \,2xy$
Let us partially differentiate the given function. The partial derivatives are
$\dfrac{{\partial f}}{{\partial x}}\, = \,2y$ ;
$\dfrac{{\partial f}}{{\partial y}}\, = \,2x$;
Therefore, we get, $\dfrac{{dy}}{{dx}}$as
$\dfrac{{dy}}{{dx}}\,\, = \,\, - \,\dfrac{{\dfrac{{\partial f}}{{\partial x}}}}{{\dfrac{{\partial f}}{{\partial y}}}}\,\, = \,\, - \dfrac{{2y}}{{2x}}\,\, = \,\, - \dfrac{y}{x}$
Hence, $2xy$ differentiated implicitly gives the result $\dfrac{{ - y}}{x}$
Additional Information:
If a function exists, some equations in $x$ and $y$ do not specifically describe $y$ as a function $x$ and cannot be easily manipulated to solve for $y$ in terms of $x$. When this happens, it implies that there is a function $y = f\left( x \right)$ that can satisfy the given equation.
We can find the derivative of $y$ with respect to $x$ using implicit differentiation without having to solve the given equation for $y$. Since we assume that $y$ can be represented as a function of $x$, the chain rule must be used if the function $y$ is distinguished.

Note: When there are several variables in a mathematical equation, partial differentiation is used to distinguish them. We find the derivative with respect to one variable only in ordinary differentiation since the function only has one variable. As a result, partial differentiation has a broader application than ordinary differentiation.