Question

What is the derivative of $$XY$$ ?

Answer 1

Hint:To find the derivative we can use the slope formula, that is $$slope={\displaystyle \frac{dy}{dx}}$$. $$dy$$ is the changes in $$Y$$and $$dx$$ is the changes in $$X$$.
 

To find the derivative of two variables in multiplication, this formula is used
$${\displaystyle \frac{d}{dx}}(uv)=u{\displaystyle \frac{d}{dx}}(v)+v{\displaystyle \frac{d}{dx}}(u)$$
Mechanically, $${\displaystyle \frac{d}{dx}}(f(x))$$ measures the rate of change of $$f(x)$$ with respect to $$x$$.Differentiation of any constant is zero. Differentiation of constant and a function is equal to constant times the differentiation of the function.

Complete step by step answer:
Let us derivate $$XY$$ with respect to $$X$$. Using the derivative formula,
$${\displaystyle \frac{d}{dx}}(uv)=u{\displaystyle \frac{d}{dx}}(v)+v{\displaystyle \frac{d}{dx}}(u)$$
Substituting the terms
$${\displaystyle \frac{d}{dX}}(XY)=X{\displaystyle \frac{d}{dX}}Y+Y{\displaystyle \frac{d}{dX}}X$$
The differentiation of $$X$$ with respect to $$X$$ is given by
$${\displaystyle \frac{d}{dX}}X=1$$
As the derivation of the function is with respect to $$X$$ , the variable $$Y$$ cannot be differentiable. $$Y$$ is not constant.
The derivative of $$XY$$ with respect to $$X$$ is
$${\displaystyle \frac{d}{dX}}(XY)=X{\displaystyle \frac{d}{dX}}Y+Y$$

Let us derivate $$XY$$ with respect to $$Y$$.Using the derivative formula,
$${\displaystyle \frac{d}{dx}}(uv)=u{\displaystyle \frac{d}{dx}}(v)+v{\displaystyle \frac{d}{dx}}(u)$$
Substituting the terms
$${\displaystyle \frac{d}{dY}}(XY)=X{\displaystyle \frac{d}{dY}}Y+Y{\displaystyle \frac{d}{dY}}X$$
The differentiation of $$Y$$ with respect to $$Y$$ is given by
$${\displaystyle \frac{d}{dY}}Y=1$$
As the derivation of the function is with respect to $$Y$$, the variable $$X$$ cannot be differentiable. $$X$$ is not constant.
The derivative of $$XY$$ with respect to $$Y$$ is
$$\therefore {\displaystyle \frac{d}{dY}}(XY)=X+Y{\displaystyle \frac{d}{dY}}X$$

Note: $${\displaystyle \frac{d}{dx}}(f(x))=\underset{h\to 0}{lim}{\displaystyle \frac{f(x+h)-f(x)}{h}}$$ is the formula for finding the derivative from the first principles. The slope is the rate of change of $$y$$ with respect to $$x$$ that means if $$x$$ is increased by an additional unit the change in $$y$$ is given by $${\displaystyle \frac{dy}{dx}}$$ . Let us understand with an example, the rate of change of displacement of an object is defined as the velocity $$Km/hr$$ that means when time is increased by one hour the displacement changes by $$Km$$. For solving derivative problems different techniques of differentiation must be known.